Open Access

Superlinear gradient system with a parameter

Boundary Value Problems20122012:110

DOI: 10.1186/1687-2770-2012-110

Received: 30 July 2012

Accepted: 25 September 2012

Published: 9 October 2012

Abstract

In this paper, we study the multiplicity of nontrivial solutions for a superlinear gradient system with saddle structure at the origin. We make use of a combination of bifurcation theory, topological linking and Morse theory.

MSC:35J10, 35J65, 58E05.

Keywords

gradient system superlinear critical group Morse theory linking

1 Introduction

In this paper, we study the existence of multiple solutions to the gradient system
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equa_HTML.gif
where Ω R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq1_HTML.gif is a bounded open domain with a smooth boundary Ω and N 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq2_HTML.gif, λ is a real parameter and A M 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq3_HTML.gif is fixed. M 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq4_HTML.gif is the set of all continuous, cooperative and symmetric matrix functions on R 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq5_HTML.gif. A matrix function A M 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq6_HTML.gif takes the form
A ( x ) = ( a ( x ) b ( x ) b ( x ) c ( x ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equb_HTML.gif

with the functions a , b , c C ( Ω ¯ , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq7_HTML.gif satisfying the conditions that b ( x ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq8_HTML.gif for all x Ω ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq9_HTML.gif, which means A is cooperative and that max x Ω ¯ { a , c } > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq10_HTML.gif.

We impose the following assumptions on the function F:

( F 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq11_HTML.gif) F C 2 ( Ω × R 2 , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq12_HTML.gif.

( F 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq13_HTML.gif) F ( x , 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq14_HTML.gif, F ( x , 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq15_HTML.gif, F z ( x , 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq16_HTML.gif for x Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq17_HTML.gif.

( F 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq18_HTML.gif) There is C > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq19_HTML.gif and 2 < p < 2 N N 2 : = 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq20_HTML.gif such that
| F ( x , z ) | C ( 1 + | z | p 1 ) , for  x Ω , z = ( u , v ) R 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equc_HTML.gif
( F 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq21_HTML.gif) There is μ > 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq22_HTML.gif, M > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq23_HTML.gif such that
0 < μ F ( x , z ) ( F ( x , z ) , z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equd_HTML.gif

for all x Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq17_HTML.gif, | z | M https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq24_HTML.gif.

( F 5 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq25_HTML.gif) F z ( x , z ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq26_HTML.gif for | z | > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq27_HTML.gif small and x Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq17_HTML.gif.

( F 6 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq28_HTML.gif) F z ( x , z ) < 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq29_HTML.gif for | z | > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq27_HTML.gif small and x Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq17_HTML.gif.

Here and in the sequel, 0 is used to denote the origin in various spaces, | | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq30_HTML.gif and ( , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq31_HTML.gif denote the norm and the inner product in R 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq5_HTML.gif, Bz denotes the matrix product in R 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq32_HTML.gif for a 2 × 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq33_HTML.gif matrix B and z = ( u , v ) R 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq34_HTML.gif. For two symmetric matrices B and C in R 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq32_HTML.gif, B > C https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq35_HTML.gif means that B C https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq36_HTML.gif is positive definite.

Let E be the Hilbert space H 0 1 ( Ω ) × H 0 1 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq37_HTML.gif endowed with the inner product
z , w = ( u , v ) , ( ϕ , ψ ) = Ω ( u ϕ + v ψ ) d x , z = ( u , v ) , w = ( ϕ , ψ ) E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Eque_HTML.gif
and the associated norm
z 2 = Ω | z | 2 d x = Ω | u | 2 + | v | 2 d x , z = ( u , v ) E . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equf_HTML.gif
By the compact Sobolev embedding E L p ( Ω ) × L p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq38_HTML.gif for p [ 2 , 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq39_HTML.gif, under the assumptions ( F 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq11_HTML.gif) and ( F 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq18_HTML.gif), the functional
Φ ( z ) = 1 2 Ω | z | 2 λ ( A ( x ) z , z ) d x Ω F ( x , z ) d x , z = ( u , v ) E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equ1_HTML.gif
(1.1)
is well defined and is of class C 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq40_HTML.gif (see [1]) with derivatives
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equg_HTML.gif

for z = ( u , v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq41_HTML.gif, w = ( ϕ , ψ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq42_HTML.gif, z 1 = ( u 1 , v 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq43_HTML.gif, z 2 = ( u 2 , v 2 ) E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq44_HTML.gif. Therefore, the solutions to (GS) λ are exactly critical points of Φ in E.

By ( F 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq13_HTML.gif) the system (GS) λ admits a trivial solution z = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq45_HTML.gif for any fixed parameter λ R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq46_HTML.gif. We are interested in finding nontrivial solutions to (GS) λ . The existence of nontrivial solutions of (GS) λ depends on the behaviors of F near zero and infinity. The purpose of this paper is to find multiple nontrivial solutions to (GS) λ with superlinear term when the trivial solution z = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq45_HTML.gif acts as a local saddle point of the energy functional Φ in the sense that the parameter λ is close to a higher eigenvalue of the linear gradient system with the given weight matrix A
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equh_HTML.gif
It is known (see [2, 3]) that for a given matrix A M 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq47_HTML.gif, ( L A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq48_HTML.gif) admits a sequence of distinct eigenvalues of finite multiplicity
0 < λ 1 A < λ 2 A < < λ k A < https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equi_HTML.gif

such that λ k A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq49_HTML.gif as k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq50_HTML.gif.

Denote by F https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq51_HTML.gif the negative part of F, i.e., F ( x , z ) = max { F ( x , z ) , 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq52_HTML.gif.

We will prove the following theorems.

Theorem 1.1 Assume ( F 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq11_HTML.gif)-( F 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq21_HTML.gif), ( F 5 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq25_HTML.gif) and let k 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq53_HTML.gif be fixed. Then there is δ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq54_HTML.gif such that when sup ( x , z ) Ω × R 2 F ( x , z ) δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq55_HTML.gif, for all λ ( λ k + 1 A δ , λ k + 1 A ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq56_HTML.gif, (GS) λ has at least three nontrivial solutions in E.

Theorem 1.2 Assume ( F 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq11_HTML.gif)-( F 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq21_HTML.gif), ( F 6 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq28_HTML.gif) and let k 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq53_HTML.gif be fixed. Then there is δ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq54_HTML.gif such that when sup ( x , z ) Ω × R 2 F ( x , z ) δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq55_HTML.gif, for all λ ( λ k + 1 A , λ k + 1 A + δ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq57_HTML.gif, (GS) λ has at least three nontrivial solutions in E.

Theorem 1.3 Assume ( F 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq11_HTML.gif)-( F 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq21_HTML.gif) and F 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq58_HTML.gif for x Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq17_HTML.gif, z R 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq59_HTML.gif with | z | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq60_HTML.gif small. Then there is δ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq54_HTML.gif such that when sup ( x , z ) Ω × R 2 F ( x , z ) δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq55_HTML.gif, for all λ ( λ k + 1 A δ , λ k + 1 A ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq61_HTML.gif, (GS) λ has at least two nontrivial solutions in E.

We give some comments and comparisons. The superlinear problems have been studied extensively via variational methods since the pioneering work of Ambrosetti and Rabinowitz [4]. Most known results on elliptic superlinear problems are contributed to a single equation with Dirichlet boundary data. Let us mention some historical progress on a single equation. When the trivial solution 0 acted as a local minimizer of the energy functional, one positive solution and one negative solution were obtained by using the mountain-pass theorem in [4] and the cut-off techniques; and a third solution was constructed in a famous paper of Wang [5] by using a two dimensional linking method and a Morse theoretic approach. When the trivial solution 0 acted as a local saddle point of the energy functional, the existence of one nontrivial solution was obtained by applying a critical point theorem, which is now well known as the generalized mountain-pass theorem, built by Rabinowitz in [6] under a global sign condition (see [1]). Some extensions were done in [7, 8]via local linking. More recently, in the work of Rabinowitz, Su and Wang [9], multiple solutions have been obtained by combining bifurcation methods, Morse theory and homological linking when 0 is a saddle point in the sense that the parameter λ is very close to a higher eigenvalue of the related linear operator.

In the current paper, we build multiplicity results for superlinear gradient systems by applying the ideas constructed in [9]. These results are new since, to the best of our knowledge, no multiplicity results for gradient systems have appeared in the literature for the case that z = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq45_HTML.gif is a saddle point of Φ.

We give some explanations regarding the conditions and conclusions. The assumptions ( F 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq11_HTML.gif)-( F 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq21_HTML.gif) are standard in the study of superlinear problems. ( F 5 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq25_HTML.gif) and ( F 6 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq28_HTML.gif) are used for bifurcation analysis. It sees that ( F 5 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq25_HTML.gif) implies that F is positive near zero, while ( F 6 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq28_HTML.gif) implies that F must be negative near zero. The local properties of F near zero are necessary for constructing homological linking. When F 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq62_HTML.gif, for any parameter λ in a bounded interval, say in [ λ k A , λ k + 1 A ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq63_HTML.gif, one can use the same arguments as in [1] to construct linking starting from λ k + 1 A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq64_HTML.gif. In our theorems, we do not require the global sign condition F 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq65_HTML.gif. When the parameter λ is close to the eigenvalue λ k + 1 A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq64_HTML.gif, the homological linking will be constructed starting from λ k + 2 A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq66_HTML.gif and this linking is different from the one in [1]. This reveals the fact that when λ is close to λ k + 1 A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq67_HTML.gif from the right-hand side, the linking starting from λ k + 2 A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq66_HTML.gif can still be constructed even if F is negative somewhere. The conditions similar to ( F 5 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq25_HTML.gif) and ( F 6 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq28_HTML.gif) were first introduced in [10] where multiple periodic solutions for the second-order Hamiltonian systems were studied via the ideas in [9]. Since we treat a different problem in the current paper, we need to present the detailed discussions although some arguments may be similar to those in [9, 10].

The paper is organized as follows. In Section 2, we collect some basic abstract tools. In Section 3, we get solutions by linking arguments and give partial estimates of homological information. In Section 4, we get solutions by bifurcation theorem and give the estimates of the Morse index. The final proofs of Theorems 1.1-1.3 are given in Section 5.

2 Preliminary

In this section, we give some preliminaries that will be used to prove the main results of the paper. We first collect some basic results on the Morse theory for a C 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq40_HTML.gif functional defined on a Hilbert space.

Let E be a Hilbert space and Φ C 2 ( E , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq68_HTML.gif. Denote K = { z E | Φ ( z ) = 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq69_HTML.gif, Φ c = { z E | Φ ( z ) c } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq70_HTML.gif, K c = { z K | Φ ( z ) = c } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq71_HTML.gif for c R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq72_HTML.gif. We say that Φ satisfies the (PS) c condition at the level c R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq73_HTML.gif if any sequence { z n } E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq74_HTML.gif satisfying Φ ( z n ) c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq75_HTML.gif, Φ ( z n ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq76_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq77_HTML.gif, has a convergent subsequence. Φ satisfies (PS) if Φ satisfies (PS) c at any c R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq72_HTML.gif.

We assume that Φ satisfies (PS) and # K < https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq78_HTML.gif. Let z 0 K https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq79_HTML.gif with Φ ( z 0 ) = c R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq80_HTML.gif and U be a neighborhood of z 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq81_HTML.gif such that U K = { z 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq82_HTML.gif. The group
C q ( Φ , z 0 ) : = H q ( Φ c U , Φ c U { z 0 } ) , q Z https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equj_HTML.gif

is called the q th critical group of Φ at z 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq81_HTML.gif, where H ( A , B ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq83_HTML.gif denotes a singular relative homology group of the pair ( A , B ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq84_HTML.gif with coefficients field F https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq85_HTML.gif (see [11, 12]).

Let a < inf Φ ( K ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq86_HTML.gif. The group
C q ( Φ , ) : = H q ( E , Φ a ) , q Z https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equk_HTML.gif

is called the q th critical group of Φ at infinity (see [13]).

We call M q : = z K dim C q ( Φ , z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq87_HTML.gif the q th Morse-type numbers of the pair ( E , Φ a ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq88_HTML.gif and β q : = dim C q ( Φ , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq89_HTML.gif the Betti numbers of the pair ( E , Φ a ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq90_HTML.gif. The core of the Morse theory [11, 12] is the following relations between M q https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq91_HTML.gif and β q https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq92_HTML.gif:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equl_HTML.gif

If K = https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq93_HTML.gif, then β q = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq94_HTML.gif for all q. Since M q β q https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq95_HTML.gif for each q Z https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq96_HTML.gif, it follows that if β q 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq97_HTML.gif for some q Z https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq98_HTML.gif, then Φ must have a critical point z https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq99_HTML.gif with C q ( Φ , z ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq100_HTML.gif. If K = { z } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq101_HTML.gif, then C q ( Φ , ) C q ( Φ , z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq102_HTML.gif for all q. Thus, if C q ( Φ , ) C q ( Φ , z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq103_HTML.gif for some q, then Φ must have a new critical point. One can use critical groups to distinguish critical points obtained by other methods and use the Morse equality to find new critical points.

For the critical groups of Φ at an isolated critical point, we have the following basic facts (see [11, 12]).

Proposition 2.1 Assume that z is an isolated critical point of Φ C 2 ( E , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq68_HTML.gif with a finite Morse index m ( z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq104_HTML.gif and nullity n ( z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq105_HTML.gif. Then
  1. (1)

    C q ( Φ , z ) δ q , m ( z ) F https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq106_HTML.gif if n ( z ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq107_HTML.gif;

     
  2. (2)

    C q ( Φ , z ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq108_HTML.gif for q [ m ( z ) , m ( z ) + n ( z ) ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq109_HTML.gif (Gromoll-Meyer [14]);

     
  3. (3)

    if C m ( z ) ( Φ , z ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq110_HTML.gif then C q ( Φ , z ) δ q , m ( z ) F https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq111_HTML.gif;

     
  4. (4)

    if C m ( z ) + n ( z ) ( Φ , z ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq112_HTML.gif then C q ( Φ , z ) δ q , m ( z ) + n ( z ) F https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq113_HTML.gif.

     

Proposition 2.2 ([15, 16])

Let 0 be an isolated critical point of Φ C 2 ( E , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq68_HTML.gif with a finite Morse index m 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq114_HTML.gif and nullity n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq115_HTML.gif. Assume that Φ has a local linking at 0 with respect to a direct sum decomposition E = E E + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq116_HTML.gif, κ = dim E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq117_HTML.gif, i.e., there exists r > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq118_HTML.gif small such that
Φ ( z ) > 0 for z E + , 0 < z r , Φ ( z ) 0 for z E , z r . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equm_HTML.gif

Then C q ( Φ , 0 ) δ q , κ Z https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq119_HTML.gif for either κ = m 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq120_HTML.gif or κ = m 0 + n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq121_HTML.gif.

The concept of local linking was introduced in [7]. In [15] a partial result was given for a C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq122_HTML.gif functional. The above result was obtained in [16].

Now, we recall an abstract linking theorem which is from [1, 12, 15].

Proposition 2.3 ([1, 12, 15])

Let E be a real Banach space with E = X Y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq123_HTML.gif and = dim X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq124_HTML.gif be finite. Suppose that Φ C 1 ( E , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq125_HTML.gif satisfies (PS) and

( Φ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq126_HTML.gif) there exist ρ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq127_HTML.gif and α > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq128_HTML.gif such that
Φ ( z ) α , z S ρ = Y B ρ , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equ2_HTML.gif
(2.1)

where B ρ = { z E | z ρ } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq129_HTML.gif,

( Φ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq130_HTML.gif) there exist R > ρ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq131_HTML.gif and e Y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq132_HTML.gif with e = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq133_HTML.gif such that
Φ ( z ) < α , z Q , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equ3_HTML.gif
(2.2)
where
Q = { z = y + s e | z R , y X , 0 s R } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equn_HTML.gif
Then Φ has a critical point z https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq99_HTML.gif with Φ ( z ) = c α https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq134_HTML.gif and
C + 1 ( Φ , z ) 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equ4_HTML.gif
(2.3)

We note here that under the framework of Proposition 2.3, S ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq135_HTML.gif and ∂Q homotopically link with respect to the direct sum decomposition E = X Y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq136_HTML.gif. S ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq135_HTML.gif and ∂Q are also homologically linked. The conclusion (2.3) follows from Theorems 1.1′ and 1.5 of Chapter II in [12]. (See also [15].)

We finally collect some properties of the eigenvalue problem ( L A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq48_HTML.gif). Associated with a matrix A M 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq47_HTML.gif, there is a compact self-adjoint operator T A : E E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq137_HTML.gif such that
T A z , w = Ω ( A ( x ) z , w ) d x , z , w E . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equo_HTML.gif
The compactness of T A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq138_HTML.gif follows from the compact embedding E L 2 ( Ω ) × L 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq139_HTML.gif. The operator T A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq138_HTML.gif possesses the property that λ A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq140_HTML.gif is an eigenvalue of ( L A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq48_HTML.gif) if and only if there is nonzero z E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq141_HTML.gif such that
λ A T A z = z . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equp_HTML.gif
( L A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq48_HTML.gif) has the sequence of distinct eigenvalues
0 < λ 1 A < λ 2 A < < λ k A < , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equq_HTML.gif
and each eigenvalue λ A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq140_HTML.gif of ( L A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq48_HTML.gif) has a finite multiplicity. For j N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq142_HTML.gif, denote
E ( λ j A ) = ker ( Δ λ j A A ) , E j = i = 1 j E ( λ i A ) , j = dim E j . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equr_HTML.gif
Set
Q λ ( z ) = Ω | z | 2 λ ( A ( x ) z , z ) d x , z E . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equs_HTML.gif
Then the following variational inequalities hold:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equt_HTML.gif

We refer to [2, 3] for more properties related to the eigenvalue problem ( L A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq48_HTML.gif) and the operator T A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq138_HTML.gif.

3 Solutions via homological linking

In this section, we give the existence a nontrivial solution of (GS) λ by applying homological linking arguments and then give some estimate of its Morse index. The following lemmas are needed.

Lemma 3.1 Assume that F satisfies ( F 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq11_HTML.gif)-( F 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq21_HTML.gif), then for any fixed λ R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq46_HTML.gif, the functional Φ satisfies the (PS) condition.

Proof By ( F 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq18_HTML.gif) and the compact embedding E L p ( Ω ) × L p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq143_HTML.gif for 1 p < 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq144_HTML.gif, it is enough to show that any sequence { z n } E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq145_HTML.gif with
| Φ ( z n ) | C , n N , Φ ( z n ) 0 , n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equ5_HTML.gif
(3.1)
is bounded in E. Here and below, we use C to denote various positive constants. We modify the arguments in [1]. Choosing a positive number β ( 1 / μ , 1 / 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq146_HTML.gif for n large, we have that
C + β z n Φ ( z n ) β Φ ( z n ) , z n . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equu_HTML.gif
By ( F 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq21_HTML.gif) we deduce that
F ( x , z ) C | z | μ C , for  z R 2 , x Ω . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equ6_HTML.gif
(3.2)
Therefore,
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equv_HTML.gif
where Λ = max x Ω ¯ A ( x ) R 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq147_HTML.gif. By the Hölder inequality and the Young inequality, we get for any ϵ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq148_HTML.gif that
z n 2 2 | Ω | μ 2 μ z n μ 2 | Ω | ( μ 2 ) μ ϵ 2 2 μ + 2 μ ϵ z n μ μ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equ7_HTML.gif
(3.3)
Thus, for a fixed ϵ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq148_HTML.gif small enough, we have by (3.3) that
C + β z n ( 1 2 β ) z n 2 + 1 2 C ( μ β 1 ) z n μ μ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equw_HTML.gif

Therefore, { z n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq149_HTML.gif is bounded in E. The proof is complete. □

Now, we construct a homological linking with respect to the direct sum decomposition of E for k 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq150_HTML.gif:
E = E k + 1 E k + 1 , dim E k + 1 = k + 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equx_HTML.gif
Take an eigenvector ϕ k + 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq151_HTML.gif corresponding to the eigenvalue λ k + 2 A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq66_HTML.gif of ( L A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq48_HTML.gif) with ϕ k + 2 = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq152_HTML.gif. Set
V k + 1 : = E k + 1 span { ϕ k + 2 } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equy_HTML.gif
Lemma 3.2 Assume that F satisfies ( F 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq11_HTML.gif)-( F 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq21_HTML.gif) and k 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq53_HTML.gif. Then there exist constants ρ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq153_HTML.gif small, α > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq128_HTML.gif such that for all λ λ : = λ k + 2 A + λ k + 1 A 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq154_HTML.gif, such that
Φ ( z ) α , for z E k + 1 with z = ρ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equ8_HTML.gif
(3.4)
Proof By the conditions ( F 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq13_HTML.gif) and ( F 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq18_HTML.gif), for ϵ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq148_HTML.gif, there is C ϵ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq155_HTML.gif such that
F ( x , z ) 1 2 S 2 ϵ | z | 2 + C ϵ | z | p , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equz_HTML.gif
where S 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq156_HTML.gif is the constant for the embedding E L 2 ( Ω ) × L 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq157_HTML.gif such that | z | 2 2 S 2 z 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq158_HTML.gif for z E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq141_HTML.gif. Since for z E k + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq159_HTML.gif,
Q λ ( z ) = Ω | z | 2 λ ( A ( x ) z , z ) d x λ k + 2 A λ λ k + 2 A z 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equaa_HTML.gif
it follows that
Φ ( z ) = 1 2 Q λ ( z ) Ω F ( x , z ) d x 1 2 λ k + 2 A λ λ k + 2 A z 2 1 2 S 2 ϵ Ω | z | 2 d x C ϵ Ω | z | p d x 1 2 λ k + 2 A λ ϵ λ k + 2 A z 2 C ˜ ϵ z p = 1 2 η ( λ , ϵ ) z 2 C ˜ ϵ z p , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equab_HTML.gif
where C ˜ ϵ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq160_HTML.gif is independent of λ and
η ( λ , ϵ ) = λ k + 2 A λ ϵ λ k + 2 A . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equ9_HTML.gif
(3.5)
Since p > 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq161_HTML.gif and the function g ( r ) = 1 2 η ( λ , ϵ ) r 2 C ˜ ϵ r p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq162_HTML.gif achieves its maximum
g max = p 2 2 p ( p C ˜ ϵ ) 2 2 p η ( λ , ϵ ) p p 2 : = α ( λ , ϵ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equ10_HTML.gif
(3.6)
on ( 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq163_HTML.gif at
ρ λ , ϵ = ( η ( λ , ϵ ) p C ˜ ϵ ) 1 p 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equ11_HTML.gif
(3.7)
we see that
Φ ( z ) α ( λ , ϵ ) , for  z E k + 1  with  z = ρ λ , ϵ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equ12_HTML.gif
(3.8)
Since η ( λ , ϵ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq164_HTML.gif is a decreasing function with respect to λ for any fixed ϵ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq148_HTML.gif small, (3.4) holds for
α : = α ( λ , ϵ ) , ρ : = ρ λ , ϵ , here  ϵ = 1 4 ( λ k + 2 A λ k + 1 A ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equac_HTML.gif

The constants α and ρ are independent of λ λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq165_HTML.gif. The proof is complete. □

Lemma 3.3 Assume that F satisfies ( F 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq11_HTML.gif), ( F 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq21_HTML.gif) and k 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq53_HTML.gif. Then there exist R > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq166_HTML.gif, δ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq54_HTML.gif and σ R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq167_HTML.gif, all independent of λ, such that when λ ( λ k + 1 A δ , λ k + 1 A + δ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq168_HTML.gif and sup ( x , z ) Ω × R 2 F ( x , z ) δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq169_HTML.gif,
Φ ( z ) σ < α , for z Q , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equ13_HTML.gif
(3.9)
where
Q = { z V k + 1 | z R , z = y + s ϕ k + 2 , y E k + 1 , s 0 } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equad_HTML.gif
Proof From ( F 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq21_HTML.gif) we deduce (3.2) with a positive constant C independent of λ. For z V k + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq170_HTML.gif, write z = y + w + ϕ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq171_HTML.gif, y E k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq172_HTML.gif, w E ( λ k + 1 A ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq173_HTML.gif, ϕ span { ϕ k + 2 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq174_HTML.gif. Assume that λ ( λ k A , λ k + 2 A ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq175_HTML.gif, then
Φ ( z ) = 1 2 Q λ ( y ) + 1 2 Q λ ( w ) + 1 2 Q λ ( ϕ ) Ω F ( x , z ) d x λ k A λ 2 λ k A y 2 + λ k + 1 A λ 2 λ k + 1 A w 2 + λ k + 2 A λ 2 λ k + 2 A ϕ 2 C z μ μ + C λ k + 2 A λ k A 2 λ k + 1 A z 2 C z μ μ + C . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equ14_HTML.gif
(3.10)
Since μ > 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq22_HTML.gif and dim V k + 1 < https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq176_HTML.gif, (3.10) shows that there exists R > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq166_HTML.gif independent of λ such that
Φ ( z ) 0 , for  z V k + 1  with  z = R . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equ15_HTML.gif
(3.11)
Now, fix such an R > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq166_HTML.gif that R > ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq177_HTML.gif with ρ given in Lemma 3.2. For y E k + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq178_HTML.gif with y R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq179_HTML.gif, we write y = w + ϕ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq180_HTML.gif, w E k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq181_HTML.gif, ϕ E ( λ k + 1 A ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq182_HTML.gif. Set Γ : = sup ( x , z ) Ω × R 2 F ( x , z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq183_HTML.gif. Then we have that
Φ ( y ) = 1 2 Q λ ( w ) + 1 2 Q λ ( ϕ ) Ω F ( x , y ) d x λ k A λ 2 λ k A w 2 + λ k + 1 A λ 2 λ k + 1 A ϕ 2 + { x Ω : F 0 } F ( x , y ) d x | λ k + 1 A λ | 2 λ k + 1 A R 2 + | Ω | Γ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equ16_HTML.gif
(3.12)
Since Q = { z = y + s ϕ k + 1 | z = R , y E k , s 0 } { y E k | y R } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq184_HTML.gif, taking
δ = ( R 2 λ k + 1 A + 2 | Ω | ) 1 α , σ = α 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equae_HTML.gif
then, when λ ( λ k + 1 A δ , λ k + 1 A + δ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq168_HTML.gif and Γ δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq185_HTML.gif,
Φ ( z ) σ < α , for  z Q . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equaf_HTML.gif

The proof is complete. □

Now, we apply Proposition 2.3 to get the following existence result with partial homological information.

Theorem 3.4 Let F satisfy ( F 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq11_HTML.gif)-( F 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq21_HTML.gif) and k 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq186_HTML.gif. Then there is δ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq54_HTML.gif such that when Γ δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq185_HTML.gif, for each λ ( λ k + 1 A δ , λ k + 1 A + δ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq168_HTML.gif, (GS) λ has one nontrivial solution z https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq187_HTML.gif with a critical group satisfying
C k + 1 + 1 ( Φ , z ) 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equ17_HTML.gif
(3.13)
Proof By Lemma 3.1, Φ satisfies (PS). By Lemmas 3.2 and 3.3, for each fixed λ ( λ k + 1 δ , λ k + 1 + δ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq188_HTML.gif, Φ satisfies ( Φ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq126_HTML.gif) and ( Φ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq130_HTML.gif) in the sense that
{ ( Φ 1 ) : Φ ( z ) α , z S k + 1 : = S ρ E k + 1 ; ( Φ 2 ) : Φ ( y ) σ < α , y Q . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equ18_HTML.gif
(3.14)

Since S k + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq189_HTML.gif and ∂Q homotopically link with respect to the decomposition E = E k + 1 E k + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq190_HTML.gif, and dim V k + 1 = k + 1 + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq191_HTML.gif, it follows from Proposition 2.3 that Φ has a critical point z E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq192_HTML.gif with positive energy Φ ( z ) α > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq193_HTML.gif and its critical group satisfying (3.13). The proof is complete. □

We give some remarks. The existence of one nontrivial solution in Theorem 3.4 is valid when F is of class C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq122_HTML.gif. From Lemma 3.2, one sees that the energy of the obtained solution is bounded away from 0 as λ is close to λ k + 1 A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq64_HTML.gif. A rough local sign condition on F is needed. If F 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq62_HTML.gif, then for any fixed λ [ λ k A , λ k + 1 A ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq194_HTML.gif, a linking with respect to E k E k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq195_HTML.gif can be constructed. Proposition 2.3 is applied again to get a nontrivial solution z https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq99_HTML.gif satisfying
C k + 1 ( Φ , z ) 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equ19_HTML.gif
(3.15)

Therefore, when a global sign condition F 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq62_HTML.gif is present, as λ is close to λ k + 1 A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq64_HTML.gif from the left-hand side, two linkings can be constructed and two nontrivial solutions can be obtained. The question is how to distinguish z https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq187_HTML.gif from z https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq99_HTML.gif. Theorem 3.4 includes the case that for λ close to λ k + 1 A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq64_HTML.gif from the right-hand side, the linking with respect to E k + 1 E k + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq196_HTML.gif is constructed provided the negative values of F are small. This phenomenon was first observed in [9].

4 Solutions via bifurcation

In this section, we get two solutions for (GS) λ via bifurcation arguments [1]. We first cite the bifurcation theorem in [1].

Proposition 4.1 (Theorem 11.35 in [1])

Let E be a Hilbert space and Ψ C 2 ( E , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq197_HTML.gif with
Ψ ( u ) = L u + H ( u ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equag_HTML.gif
where L L ( E , E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq198_HTML.gif is symmetric and H ( u ) = o ( u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq199_HTML.gif as u 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq200_HTML.gif. Consider the equation
L u + H ( u ) = λ u . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equ20_HTML.gif
(4.1)
Let μ σ ( L ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq201_HTML.gif be an isolated eigenvalue of finite multiplicity. Then either
  1. (i)

    ( μ , 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq202_HTML.gif is not an isolated solution of (4.1) in { μ } × E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq203_HTML.gif, or

     
  2. (ii)

    there is a one-sided neighborhood Λ of μ such that for all λ Λ { μ } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq204_HTML.gif, (4.1) has at least two distinct nontrivial solutions, or

     
  3. (iii)

    there is a neighborhood Λ of μ such that for all λ Λ { μ } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq204_HTML.gif, (4.1) has at least one nontrivial solution.

     

We apply Proposition 4.1 to get two nontrivial solutions of (GS) λ for λ close to an eigenvalue of ( L A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq48_HTML.gif) and then give the estimates of the Morse index.

Theorem 4.2 Assume that F satisfies ( F 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq11_HTML.gif)-( F 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq18_HTML.gif). Let k 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq53_HTML.gif be fixed. Then there exists δ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq205_HTML.gif such that (GS) λ has at least two nontrivial solutions for
  1. (1)

    every λ ( λ k + 1 A δ , λ k + 1 A ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq206_HTML.gif if ( F 5 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq25_HTML.gif) holds;

     
  2. (2)

    every λ ( λ k + 1 A , λ k + 1 A + δ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq207_HTML.gif if ( F 6 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq28_HTML.gif) holds.

     
Furthermore, the Morse index m ( z λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq208_HTML.gif and the nullity n ( z λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq209_HTML.gif of such a solution z λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq210_HTML.gif satisfy
k m ( z λ ) m ( z λ ) + n ( z λ ) k + 1 , for 0 < | λ λ k + 1 A | < δ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equ21_HTML.gif
(4.2)

Proof Under the assumptions ( F 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq11_HTML.gif)-( F 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq13_HTML.gif), for each eigenvalue λ j A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq211_HTML.gif of ( L A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq48_HTML.gif), ( λ j A , 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq212_HTML.gif is a bifurcation point of (GS) λ (see [1]).

Let ( λ , z ) R × E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq213_HTML.gif be a solution of (GS) λ near ( λ k + 1 A , 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq214_HTML.gif which satisfies
{ Δ z = λ A ( x ) z + F ( x , z ) , x Ω , z = ( u , v ) = 0 , x Ω . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equ22_HTML.gif
(4.3)
By ( F 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq11_HTML.gif) and ( F 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq13_HTML.gif), we have
F ( x , z ) = F ( x , 0 ) + F z ( t , η z ) x = F z ( x , η z ) z , for some  0 < η < 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equ23_HTML.gif
(4.4)
Let ( F 5 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq25_HTML.gif) hold. By the elliptic regularity theory (see [17]), z > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq215_HTML.gif small implies z C > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq216_HTML.gif small. Then by ( F 5 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq25_HTML.gif), we have that
F z ( z , η z ( x ) ) > 0 , x Ω . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equ24_HTML.gif
(4.5)
Now, consider the linear eigenvalue gradient system:
{ Δ y F z ( x , η z ( x ) ) y = μ A ( x ) y , x Ω , y = 0 , x Ω . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equ25_HTML.gif
(4.6)

We denote the distinct eigenvalues of (4.6) by μ 1 ( z ) < μ 2 ( z ) < < μ i ( z ) < https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq217_HTML.gif as z 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq218_HTML.gif. By ( F 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq13_HTML.gif), if we take z = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq45_HTML.gif, then for each i N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq219_HTML.gif, there is j N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq220_HTML.gif such that μ i ( 0 ) = λ j A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq221_HTML.gif. By (4.5), the standard variational characterization of the eigenvalues of (4.6) shows that μ i ( z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq222_HTML.gif is less than the corresponding j th ordered eigenvalue λ j A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq211_HTML.gif of ( L A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq48_HTML.gif). Furthermore, μ i ( z ) λ j A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq223_HTML.gif as z 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq224_HTML.gif in E. By (4.3) and (4.4), we see that z is a solution of (4.6) with eigenvalue λ. It must be that λ < λ k + 1 A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq225_HTML.gif since λ is close to λ k + 1 A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq64_HTML.gif. Therefore, the case (ii) of Proposition 4.1 occurs under the given conditions. This proves the case (1). The existence for the case (2) is proved in the same way.

Now, we estimate the Morse indices for the solutions obtained above. Let z λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq210_HTML.gif be a bifurcation solution of (GS) λ . Then
z λ 0 as  λ λ k + 1 A . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equah_HTML.gif
Applying the elliptic regularity theory, we have that
z λ C 0 , λ λ k + 1 A . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equ26_HTML.gif
(4.7)
For each y E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq226_HTML.gif, we have
Φ ( z λ ) y , y = Q λ ( y ) Ω ( F z ( x , z λ ) y , y ) d x . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equai_HTML.gif
Therefore, for y E k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq227_HTML.gif,
Φ ( z λ ) y , y λ k A λ λ k A y 2 + Ω | ( F z ( x , z λ ) y , y ) | d x , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equaj_HTML.gif
and for ϕ E k + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq228_HTML.gif,
Φ ( z λ ) ϕ , ϕ λ k + 2 A λ λ k + 2 A ϕ 2 Ω | ( F z ( x , z λ ) ϕ , ϕ ) | d x . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equak_HTML.gif
By ( F 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq13_HTML.gif) and (4.7), there exists δ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq54_HTML.gif such that when 0 < | λ λ k + 1 A | < δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq229_HTML.gif,
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equal_HTML.gif

Therefore, the Morse index m ( z λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq208_HTML.gif and the nullity n ( z λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq209_HTML.gif of z λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq210_HTML.gif satisfy (4.2). The proof is complete. □

5 Proofs of main theorems

In this section, we give the proof of main theorems in this paper. We first compute the critical groups of Φ at both infinity and zero.

Lemma 5.1 (see [5])

Let F satisfy ( F 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq11_HTML.gif)-( F 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq21_HTML.gif), then for any fixed λ R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq46_HTML.gif,
C q ( Φ , ) 0 , for all q Z . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equ27_HTML.gif
(5.1)
Proof The idea of the proof comes from the famous paper [5]. We include a sketched proof in an abstract version. Given λ R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq46_HTML.gif, denote B 1 = { z E : z 1 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq230_HTML.gif, S 1 = B 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq231_HTML.gif. Modifying the arguments in [5], we get the following facts:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equ28_HTML.gif
(5.2)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equ29_HTML.gif
(5.3)
The following arguments are from [10]. As Φ ( 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq232_HTML.gif, it follows from (5.2) and (5.3) that for each z S 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq233_HTML.gif, there is a unique τ ( z ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq234_HTML.gif such that
Φ ( τ ( z ) z ) = a . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equ30_HTML.gif
(5.4)
By (5.4) and the implicit function theorem, we have that τ C ( S 1 , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq235_HTML.gif. Define
π ( z ) = { 1 , if  Φ ( z ) a , z 1 τ ( z 1 z ) , if  Φ ( z ) > a , z 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equam_HTML.gif
Then π C ( E { 0 } , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq236_HTML.gif. Define a map ϱ : [ 0 , 1 ] × E { 0 } E { 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq237_HTML.gif by
ϱ ( t , z ) = ( 1 t ) z + t π ( z ) z . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equ31_HTML.gif
(5.5)
Clearly, ϱ is continuous, and for all z E { 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq238_HTML.gif with Φ ( z ) > a https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq239_HTML.gif, by (5.4),
Φ ( ϱ ( 1 , z ) ) = Φ ( π ( z 1 z ) z 1 z ) = a . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equan_HTML.gif
Therefore,
ϱ ( 1 , z ) Φ a for all  z E { 0 } , ϱ ( t , z ) = z for all  t [ 0 , 1 ] , z Φ a , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equao_HTML.gif
and so Φ a https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq240_HTML.gif is a strong deformation retract of E { 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq241_HTML.gif. Hence,
C q ( Φ , ) : = H q ( E , Φ a ) H q ( E , E { 0 } ) H q ( B 1 , S 1 ) 0 , q Z https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equap_HTML.gif

since S 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq242_HTML.gif is contractible, which follows from the fact that dim E = https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq243_HTML.gif. □

Lemma 5.2 Let F satisfy ( F 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq11_HTML.gif)-( F 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq18_HTML.gif).
  1. (1)

    For λ ( λ k A , λ k + 1 A ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq244_HTML.gif, C q ( Φ , 0 ) δ q , k F https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq245_HTML.gif.

     
  2. (2)

    For λ ( λ k + 1 A , λ k + 2 A ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq246_HTML.gif, C q ( Φ , 0 ) δ q , k + 1 F https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq247_HTML.gif.

     
  3. (3)

    For λ = λ k + 1 A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq248_HTML.gif, if F ( x , z ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq249_HTML.gif for | z | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq60_HTML.gif small, then C q ( Φ , 0 ) δ q , k F https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq250_HTML.gif.

     
  4. (4)

    For λ = λ k + 1 A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq248_HTML.gif, if F ( x , z ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq251_HTML.gif for | z | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq60_HTML.gif small, then C q ( Φ , 0 ) δ q , k + 1 F https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq252_HTML.gif.

     
Proof By ( F 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq13_HTML.gif), we have
Φ ( 0 ) y , y = Q λ ( y ) = Ω | y | 2 λ ( A ( x ) y , y ) d x , y E . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equaq_HTML.gif
  1. (1)

    When λ ( λ k A , λ k + 1 A ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq244_HTML.gif, z = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq45_HTML.gif is a nondegenerate critical point of Φ with the Morse index m 0 = k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq253_HTML.gif, thus C q ( Φ , 0 ) δ q , k F https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq254_HTML.gif.

     
  2. (2)

    When λ ( λ k + 1 A , λ k + 2 A ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq246_HTML.gif, z = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq45_HTML.gif is a nondegenerate critical point of Φ with the Morse index m 0 = k + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq255_HTML.gif, thus C q ( Φ , 0 ) δ q , k + 1 F https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq256_HTML.gif.

     
  3. (3)

    When λ = λ k + 1 A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq248_HTML.gif, z = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq45_HTML.gif is a degenerate critical point of Φ with the Morse index m 0 = k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq257_HTML.gif and the nullity n 0 = dim E ( λ k + 1 A ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq258_HTML.gif, m 0 + n 0 = k + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq259_HTML.gif.

     

Assume that F ( x , z ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq249_HTML.gif for | z | σ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq260_HTML.gif with σ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq261_HTML.gif small. We will show that Φ has a local linking structure at z = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq45_HTML.gif with respect to E = E k E k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq262_HTML.gif. If this has been done, then by Proposition 2.2, we have C q ( Φ , 0 ) δ q , k F https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq263_HTML.gif.

Now, Φ can be written as
Φ ( z ) = 1 2 Q λ k + 1 A ( z ) Ω F ( x , z ) d x . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equar_HTML.gif
By ( F 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq13_HTML.gif) and ( F 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq18_HTML.gif), for ϵ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq148_HTML.gif, there is C ϵ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq155_HTML.gif such that
F ( x , z ) 1 2 ϵ | z | 2 + C ϵ | z | p , z R 2 , x Ω . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equas_HTML.gif
Hence, for z E k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq264_HTML.gif, we have that
Φ ( z ) λ k A λ k + 1 A 2 λ k A z 2 + ϵ 2 | z | 2 2 + C ϵ | z | p p . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equat_HTML.gif
Since E k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq265_HTML.gif is finite dimensional, all norms on E k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq265_HTML.gif are equivalent, hence for r > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq118_HTML.gif small,
Φ ( z ) 0 , for  z E k , z r . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equau_HTML.gif
By ( F 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq18_HTML.gif), we have that for some C > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq19_HTML.gif,
| F ( x , z ) | C | z | p , | z | σ , x Ω . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equ32_HTML.gif
(5.6)
For z E k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq266_HTML.gif, we write z = y + w https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq267_HTML.gif where y E ( λ k + 1 A ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq268_HTML.gif and w E k + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq269_HTML.gif. Then
Φ ( z ) λ k + 2 A λ k + 1 A 2 λ k + 2 A w 2 Ω F ( x , z ) d x . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equ33_HTML.gif
(5.7)
For x Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq17_HTML.gif with | z ( x ) | σ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq270_HTML.gif, we have | w ( x ) | 2 3 | z ( x ) | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq271_HTML.gif. Hence, by (5.6) and the Poincaré inequality, we have for various constants C > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq19_HTML.gif,
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equ34_HTML.gif
(5.8)
For x Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq17_HTML.gif with | z ( x ) | σ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq272_HTML.gif, F ( x , z ( x ) ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq273_HTML.gif. Therefore,
Φ ( z ) λ k + 2 A λ k + 1 A 2 λ k + 2 A w 2 { x : | z ( x ) | σ } F ( x , z ( x ) ) d x C w p λ k + 2 A λ k + 1 A 2 λ k + 2 A w 2 C w p . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equ35_HTML.gif
(5.9)
Since p > 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq161_HTML.gif, for r > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq118_HTML.gif small,
Φ ( z ) > 0 , for  z = y + w  with  w 0 , z r . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equ36_HTML.gif
(5.10)
For z = y E ( λ k + 1 A ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq274_HTML.gif, it must hold that
Φ ( y ) = Ω F ( x , y ( x ) ) d x > 0 if  0 < y r . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equ37_HTML.gif
(5.11)
Here we use a potential convention that (GS) λ has finitely many solutions and then 0 is isolated. Otherwise, one would have that as r > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq118_HTML.gif small, y r https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq275_HTML.gif implies | y ( x ) | δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq276_HTML.gif for all x Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq17_HTML.gif, F ( x , y ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq277_HTML.gif for all x Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq17_HTML.gif. Thus, 0 would not be an isolated critical point of Φ and (GS) λ would have infinitely many nontrivial solutions. By (5.10) and (5.11), we verify that
Φ ( z ) > 0 , for  z E k , 0 < z r . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equav_HTML.gif
Applying Proposition 2.2, we obtain
C q ( Φ , 0 ) = δ q , k F . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equaw_HTML.gif

(4) When F ( x , z ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq251_HTML.gif for | z | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq60_HTML.gif small, a similar argument shows that Φ has a local linking structure at z = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq45_HTML.gif with respect to E = E k + 1 E k + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq278_HTML.gif. By Proposition 2.2, it follows that C q ( Φ , 0 ) δ q , k + 1 F https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq279_HTML.gif. □

Finally, we prove the theorems.

Proof of Theorem 1.1 It follows from ( F 5 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq25_HTML.gif) that F ( x , z ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq251_HTML.gif for | z | > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq27_HTML.gif small. By Theorem 3.4 for the part λ ( λ k + 1 A δ , λ k + 1 A ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq280_HTML.gif, (GS) λ has a nontrivial solution z λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq281_HTML.gif satisfying
C k + 1 + 1 ( Φ , z λ ) 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equ38_HTML.gif
(5.12)
By Theorem 4.2(1), (GS) λ has two nontrivial solutions z λ i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq282_HTML.gif ( i = 1 , 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq283_HTML.gif) with their Morse indices satisfying
k m ( z λ i ) m ( z λ i ) + n ( z λ i ) k + 1 , i = 1 , 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equax_HTML.gif
From Proposition 2.1(2), we have that
C q ( Φ , z λ i ) 0 , q [ k , k + 1 ] , i = 1 , 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equ39_HTML.gif
(5.13)

From (5.12) and (5.13), we see that z λ z λ i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq284_HTML.gif ( i = 1 , 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq283_HTML.gif). The proof is complete. □

Proof of Theorem 1.2 With the same argument as above, it follows from Theorem 4.2(2) and Theorem 3.4 for the part λ ( λ k + 1 A , λ k + 1 A + δ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq285_HTML.gif. We omit the details. □

Proof of Theorem 1.3 By Theorem 3.4 for the part λ ( λ k + 1 A δ , λ k + 1 A ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq286_HTML.gif, (GS) λ has a solution z λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq281_HTML.gif with its energy Φ ( z λ ) α > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq287_HTML.gif and
C k + 1 + 1 ( Φ , z λ ) 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equ40_HTML.gif
(5.14)
By Lemma 5.1 and Lemma 5.2(3), we have that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equ41_HTML.gif
(5.15)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equ42_HTML.gif
(5.16)
Assume that (GS) λ has only two solutions 0 and z λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq281_HTML.gif. Choose a , b R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq288_HTML.gif such that a < 0 < b < Φ ( z λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq289_HTML.gif. Then by the deformation and excision properties of singular homology (see [12]), we have
{ C q ( Φ , ) H q ( E , Φ a ) ; C q ( Φ , 0 ) H q ( Φ b , Φ a ) ; C q ( Φ , z λ ) H q ( E , Φ b ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equ43_HTML.gif
(5.17)
By (5.17), the long exact sequences for the topological triple ( E , Φ b , Φ a ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq290_HTML.gif read as
C q + 1 ( Φ , ) C q + 1 ( Φ , z λ ) C q ( Φ , 0 ) C q ( Φ , ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equ44_HTML.gif
(5.18)
We deduce by (5.15) and (5.18) that
C q + 1 ( Φ , z λ ) C q ( Φ , 0 ) , for  q Z . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equ45_HTML.gif
(5.19)
Take q = k + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq291_HTML.gif in (5.19), then
C k + 1 + 1 ( Φ , z λ ) C k + 1 ( Φ , 0 ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_Equay_HTML.gif

which contradicts (5.14). The proof is complete. □

We finally remark that Theorem 1.1 is valid for λ ( λ 1 A δ , λ 1 A ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq292_HTML.gif, from which one sees that z = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-110/MediaObjects/13661_2012_Article_241_IEq45_HTML.gif is a local minimizer of Φ.

Declarations

Acknowledgements

The authors are grateful to the anonymous referee for his/her valuable suggestions. The second author was supported by NSFC11271264, NSFC11171204, KZ201010028027 and PHR201106118.

Authors’ Affiliations

(1)
School of Mathematical Sciences, Capital Normal University

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