Existence of solutions for a class of biharmonic equations with the Navier boundary value condition

Boundary Value Problems20122012:154

DOI: 10.1186/1687-2770-2012-154

Received: 18 July 2012

Accepted: 14 December 2012

Published: 28 December 2012

Abstract

In this paper, the existence of at least one nontrivial solution for a class of fourth-order elliptic equations with the Navier boundary value conditions is established by using the linking methods.

Keywords

biharmonic Navier boundary value problems local linking

1 Introduction

Consider the following Navier boundary value problem:
{ 2 u ( x ) + l u = f ( x , u ) , in  Ω ; u = u = 0 on  Ω , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_Equ1_HTML.gif
(1.1)

where 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq1_HTML.gif is the biharmonic operator, l R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq2_HTML.gif and Ω is a bounded smooth domain in http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq3_HTML.gif ( N > 4 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq4_HTML.gif).

The conditions imposed on f ( x , t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq5_HTML.gif are as follows:

(H1) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq6_HTML.gif , and there are constants C 1 , C 2 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq7_HTML.gif such that
| f ( x , t ) | C 1 + C 2 | t | s 1 , x Ω , t R , s ( 2 , p ) ( N > 4 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_Equa_HTML.gif

where p = 2 N N 4 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq8_HTML.gif;

(H2) f ( x , t ) = ( | t | ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq9_HTML.gif, | t | 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq10_HTML.gif, uniformly on Ω;

(H3) lim | t | f ( x , t ) t = + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq11_HTML.gif uniformly on Ω;

(H4) There is a constant θ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq12_HTML.gif such that for all ( x , t ) Ω × R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq13_HTML.gif and s [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq14_HTML.gif,
θ ( f ( x , t ) t 2 F ( x , t ) ) ( s f ( x , s t ) t 2 F ( x , s t ) ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_Equb_HTML.gif

where F ( x , t ) = 0 t f ( x , s ) d s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq15_HTML.gif;

(H5) For some δ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq16_HTML.gif, either
F ( x , t ) 0 , for  | t | δ , x Ω , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_Equc_HTML.gif
or
F ( x , t ) 0 , for  | t | δ , x Ω . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_Equd_HTML.gif

This fourth-order semilinear elliptic problem has been studied by many authors. In [1], there was a survey of results obtained in this direction. In [2], Micheletti and Pistoia showed that (1.1) admits at least two solutions by a variation of linking if f ( x , u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq17_HTML.gif is sublinear. And in [3], the authors proved that the problem (1.1) has at least three solutions by a variational reduction method and a degree argument. In [4], Zhang and Li showed that (1.1) admits at least two nontrivial solutions by the Morse theory and local linking if f ( x , u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq17_HTML.gif is superlinear and subcritical on u. In [5], Zhang and Wei obtained the existence of infinitely many solutions for the problem (1.1) where the nonlinearity involves a combination of superlinear and asymptotically linear terms. As far as the problem (1.1) is concerned, existence results of sign-changing solutions were also obtained. See, e.g., [6, 7] and the references therein.

We will use linking methods to give the existence of at least one nontrivial solution for (1.1).

Let X be a Banach space with a direct sum decomposition
X = X 1 X 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_Eque_HTML.gif
The function I C 1 ( X , R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq18_HTML.gif has a local linking at 0, with respect to ( X 1 , X 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq19_HTML.gif if for some r > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq20_HTML.gif,
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_Equ2_HTML.gif
(1.2)
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_Equ3_HTML.gif
(1.3)

It is clear that 0 is a critical point of I.

The notion of local linking generalizes the notions of local minimum and local maximum. When 0 is a non-degenerate critical point of a functional of class C 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq21_HTML.gif defined on a Hilbert space and I ( 0 ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq22_HTML.gif, I has local linking at 0.

The condition of local linking was introduced in [8] under stronger assumptions
I ( u ) c > 0 , u X 1 , u = r , dim X 2 < . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_Equf_HTML.gif
Under those assumptions, the existence of nontrivial critical points was proved for functionals which are
  1. (a)

    bounded below [8],

     
  2. (b)

    superquadratic [8] and

     
  3. (c)

    asymptotically quadratic [9].

     

The cases (a), (b) and (c) were considered in [10] with only conditions (1.2) and (1.3), and three theorems about critical points were proved. Using these theorems, the author in [10] proved the existence of at least one nontrivial solution for the second-order elliptic boundary value problem with the Dirichlet boundary value condition. In the present paper, we also use the three theorems in [10] and the linking technique to give the existence of at least one nontrivial solution for the biharmonic problem (1.1) under a weaker condition. The main results are as follows.

Theorem 1.1 Assume the conditions (H1)-(H4) hold. If l is an eigenvalue of −△ (with the Dirichlet boundary condition), assume also (H5). Then the problem (1.1) has at least one nontrivial solution.

We also consider asymptotically quadratic functions.

Let 0 < λ 1 < λ 2 < < λ k < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq23_HTML.gif be the eigenvalues of ( , H 0 1 ( Ω ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq24_HTML.gif. Then μ j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq25_HTML.gif ( j N + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq26_HTML.gif) is the eigenvalue of ( 2 + l , H 2 ( Ω ) H 0 1 ( Ω ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq27_HTML.gif, where μ j = λ j ( λ j l ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq28_HTML.gif. We assume that

(H6) f ( x , u ) = f u + ( | u | ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq29_HTML.gif, | u | http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq30_HTML.gif, uniformly in Ω, and μ k < f < μ k + 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq31_HTML.gif.

Theorem 1.2 Assume the conditions (H1), (H6) and one of the following conditions:

(A1) λ j < l < λ j + 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq32_HTML.gif, j k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq33_HTML.gif;

(A2) λ j = l < λ j + 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq34_HTML.gif, j k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq33_HTML.gif, for some δ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq16_HTML.gif,
F ( x , u ) 0 , for | u | > δ , x Ω ; http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_Equg_HTML.gif
(A3) λ j < l = λ j + 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq35_HTML.gif, j k http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq33_HTML.gif, for some δ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq16_HTML.gif,
F ( x , u ) 0 , for | u | δ , x Ω . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_Equh_HTML.gif

Then the problem (1.1) has at least one nontrivial solution.

2 Preliminaries

Let X be a Banach space with a direct sum decomposition
X = X 1 X 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_Equi_HTML.gif
Consider two sequences of a subspace:
X 0 1 X 1 1 X 1 , X 0 2 X 1 2 X 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_Equj_HTML.gif
such that
X j = n N X n j , j = 1 , 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_Equk_HTML.gif
For every multi-index α = ( α 1 , α 2 ) N 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq36_HTML.gif, let X α = X α 1 X α 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq37_HTML.gif. We know that
α β α 1 β 1 , α 2 β 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_Equl_HTML.gif

A sequence ( α n ) N 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq38_HTML.gif is admissible if for every α N 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq39_HTML.gif, there is m N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq40_HTML.gif such that n m α n α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq41_HTML.gif. For every I : X R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq42_HTML.gif, we denote by I α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq43_HTML.gif the function I restricted X α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq44_HTML.gif.

Definition 2.1 Let I be locally Lipschitz on X and c R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq45_HTML.gif. The functional I satisfies the ( C ) c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq46_HTML.gif condition if every sequence ( u α n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq47_HTML.gif such that ( α n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq48_HTML.gif is admissible and
u α n X α n , I ( u α n ) c , ( 1 + u α n ) I ( u α n ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_Equm_HTML.gif

contains a subsequence which converges to a critical point of I.

Definition 2.2 Let I be locally Lipschitz on X and c R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq45_HTML.gif. The functional I satisfies the ( C ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq49_HTML.gif condition if every sequence ( u α n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq47_HTML.gif such that ( α n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq48_HTML.gif is admissible and
u α n X α n , sup n I ( u α n ) c , ( 1 + u α n ) I ( u α n ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_Equn_HTML.gif

contains a subsequence which converges to a critical point of I.

Remark 2.1
  1. 1.

    The ( C ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq49_HTML.gif condition implies the ( C ) c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq46_HTML.gif condition for every c R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq45_HTML.gif.

     
  2. 2.

    When the ( C ) c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq46_HTML.gif sequence is bounded, then the sequence is a ( P S ) c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq50_HTML.gif sequence (see [11]).

     
  3. 3.
    Without loss of generality, we assume that the norm in X satisfies
    u 1 + u 2 2 = u 1 2 + u 2 2 , u j X j , j = 1 , 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_Equo_HTML.gif
     
Definition 2.3 Let X be a Banach space with a direct sum decomposition
X = X 1 X 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_Equp_HTML.gif
The function I C 1 ( X , R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq18_HTML.gif has a local linking at 0, with respect to ( X 1 , X 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq51_HTML.gif, if for some r > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq20_HTML.gif,
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_Equq_HTML.gif

Lemma 2.1 (see [10])

Suppose that I C 1 ( X , R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq18_HTML.gif satisfies the following assumptions:

(B1) I has a local linking at 0 and X 1 { 0 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq52_HTML.gif;

(B2) I satisfies ( P S ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq53_HTML.gif;

(B3) I maps bounded sets into bounded sets;

(B4) for every m N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq54_HTML.gif, I ( u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq55_HTML.gif, u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq56_HTML.gif, u X = X m 1 X 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq57_HTML.gif. Then I has at least two critical points.

Remark 2.2 Assume I satisfies the ( C ) c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq46_HTML.gif condition. Then this theorem still holds.

Let X be a real Hilbert space and let I C 1 ( X , R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq18_HTML.gif. The gradient of I has the form
I ( u ) = A u + B ( u ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_Equr_HTML.gif

where A is a bounded self-adjoint operator, 0 is not the essential spectrum of A, and B is a nonlinear compact mapping.

We assume that there exist an orthogonal decomposition,
X = X 1 + X 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_Equs_HTML.gif
and two sequences of finite-dimensional subspaces,
X 0 1 X 1 1 X 1 1 X 1 , X 0 2 X 1 2 X 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_Equt_HTML.gif
such that
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_Equu_HTML.gif
For every multi-index α = ( α 1 , α 2 ) N 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq36_HTML.gif, we denote by X α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq44_HTML.gif the space
X α 1 X α 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_Equv_HTML.gif

by p α : X X α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq58_HTML.gif the orthogonal projector onto X α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq44_HTML.gif, and by M ( L ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq59_HTML.gif the Morse index of a self-adjoint operator L.

Lemma 2.2 (see [10])

I satisfies the following assumptions:
  1. (i)

    I has a local linking at 0 with respect to ( X 1 , X 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq51_HTML.gif;

     
  2. (ii)
    there exists a compact self-adjoint operator B http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq60_HTML.gif such that
    B ( u ) = B ( u ) + ( u ) , u ; http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_Equw_HTML.gif
     
  3. (iii)

    A + B http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq61_HTML.gif is invertible;

     
  4. (iv)
    for infinitely many multiple-indices α : = ( n , n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq62_HTML.gif,
    M ( ( A + P α B ) | X α ) dim X n 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_Equx_HTML.gif
     

Then I has at least two critical points.

3 The proof of main results

Proof of Theorem 1.1 (1) We shall apply Lemma 2.1 to the functional
I ( u ) = 1 2 Ω ( | u | 2 l | u | 2 ) d x Ω F ( x , u ) d x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_Equy_HTML.gif
defined on X = H 0 1 ( Ω ) H 2 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq63_HTML.gif. We consider only the case l = λ j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq64_HTML.gif, and
F ( x , u ) 0 , for  | u | δ , x Ω . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_Equ4_HTML.gif
(3.1)

Then other case is similar and simple.

Let X 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq65_HTML.gif be the finite dimensional space spanned by the eigenfunctions corresponding to negative eigenvalues of 2 + l http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq66_HTML.gif and let X 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq67_HTML.gif be its orthogonal complement in X. Choose a Hilbertian basis e n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq68_HTML.gif ( n 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq69_HTML.gif) for X and define
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_Equz_HTML.gif

By the condition (H1) and Sobolev inequalities, it is easy to see that the functional I belongs to C 1 ( X , R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq70_HTML.gif and maps bounded sets to bounded sets.

(2) We claim that I has a local linking at 0 with respect to ( X 1 , X 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq51_HTML.gif. Decompose X 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq67_HTML.gif into V + W http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq71_HTML.gif when V = ker ( 2 + l ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq72_HTML.gif, W = ( X 2 + V ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq73_HTML.gif. Also, set u = v + w http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq74_HTML.gif, u X 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq75_HTML.gif, v V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq76_HTML.gif, w W http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq77_HTML.gif. By the equivalence of norm in the finite-dimensional space, there exists C > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq78_HTML.gif such that
v C v X , v V . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_Equ5_HTML.gif
(3.2)
It follows from (H1) and (H2) that for any ϵ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq79_HTML.gif, there exists C ϵ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq80_HTML.gif such that
| F ( x , u ) | ϵ u 2 + C ϵ | u | s . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_Equ6_HTML.gif
(3.3)
Hence, we obtain
I ( u ) 1 2 Ω ( | u | 2 l | u | 2 ) d x + ϵ Ω u 2 d x + c u X s + 1 m u 2 + ϵ Ω u 2 d x + c u X s + 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_Equaa_HTML.gif
where m > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq81_HTML.gif, c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq82_HTML.gif is a constant and hence, for r > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq20_HTML.gif small enough,
I ( u ) 0 , u X 2 , u X r . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_Equab_HTML.gif
Let u = v + w X 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq83_HTML.gif be such that u X r 1 = δ 2 C http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq84_HTML.gif and let
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_Equac_HTML.gif
From (3.2), we have
| v ( x ) | v C v δ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_Equad_HTML.gif
for all u r 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq85_HTML.gif and x Ω http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq86_HTML.gif. On the one hand, one has | u ( x ) | | v ( x ) | + | w ( x ) | v + δ 2 δ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq87_HTML.gif for all x Ω 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq88_HTML.gif. Hence, from (H5), we obtain
Ω 1 F ( x , u ) d x 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_Equae_HTML.gif
On the other hand, we have
| u ( x ) | | v ( x ) | + | w ( x ) | δ 2 + | w ( x ) | 2 | w ( x ) | http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_Equaf_HTML.gif
for all x Ω 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq89_HTML.gif. It follows from (3.3) that
F ( x , u ) ϵ u 2 + C ϵ | u | s + 1 4 ϵ w 2 + 2 s + 1 C ϵ | w | s + 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_Equag_HTML.gif
for all x Ω 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq90_HTML.gif and all u X 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq91_HTML.gif with u r 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq85_HTML.gif, which implies that
Ω F ( x , u ) d x 4 ϵ Ω 2 w 2 d x + Ω 2 2 s + 1 C ϵ | w | s + 1 d x 4 ( C 3 ) 2 ϵ w 2 + ( 2 C 3 ) λ + 1 C ϵ w s + 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_Equah_HTML.gif
where C 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq92_HTML.gif is a constant. Hence, there exist positive constants C http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq93_HTML.gif, C 4 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq94_HTML.gif and C 5 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq95_HTML.gif such that
I ( u ) = 1 2 w 2 1 2 Ω l | w | 2 d x Ω 2 F ( x , u ) d x Ω 1 F ( x , u ) d x C w 2 4 ( C 3 ) 2 ϵ w 2 ( 2 C 3 ) λ + 1 C ϵ w s + 1 Ω 1 G ( x , u ) d x C 4 w 2 C 5 w s + 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_Equai_HTML.gif
for all u X 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq75_HTML.gif with u r 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq85_HTML.gif, which implies that
I ( u ) 0 , u X 1  with  u r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_Equaj_HTML.gif

for 0 < r http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq96_HTML.gif small enough.

(3) We claim that I satisfies ( C ) c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq46_HTML.gif. Consider a sequence ( u α n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq47_HTML.gif such that ( u α n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq47_HTML.gif is admissible and
u α n X α n , I ( u α n ) c , ( 1 + u α n ) I ( u α n ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_Equ7_HTML.gif
(3.4)
and
lim n Ω ( 1 2 f ( x , u α n ) u α n F ( x , u α n ) ) d x = c . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_Equ8_HTML.gif
(3.5)
Let w α n = u α n 1 u α n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq97_HTML.gif. Up to a subsequence, we have
w α n w in  X , w α n w in  L 2 , w α n ( x ) w ( x ) a.e.  x Ω . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_Equak_HTML.gif
If w = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq98_HTML.gif, we choose a sequence { t n } [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq99_HTML.gif such that
I ( t n u α n ) = max t [ 0 , 1 ] I ( t u α n ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_Equal_HTML.gif
For any m > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq81_HTML.gif, let v α n = 2 m w α n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq100_HTML.gif. By the Sobolev imbedded theory, we have
lim n Ω F ( x , v α n ) d x = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_Equam_HTML.gif
So, for n large enough, 2 m u α n 1 ( 0 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq101_HTML.gif, and combining Ehrling-Nirenberg-Gagliardo inequality, we have
I ( t n u α n ) I ( v α n ) m ϵ m 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_Equ9_HTML.gif
(3.6)

where ϵ is a small enough constant.

That is, I ( t n u α n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq102_HTML.gif. Now, I ( 0 ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq22_HTML.gif, I ( u α n ) c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq103_HTML.gif, we know that t n [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq104_HTML.gif and
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_Equ10_HTML.gif
(3.7)
Therefore, using (H3), we have
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_Equan_HTML.gif

This contradicts (3.5).

If w 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq105_HTML.gif, then the set = { x Ω : w ( x ) 0 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq106_HTML.gif has a positive Lebesgue measure. For x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq107_HTML.gif, we have | u α n ( x ) | http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq108_HTML.gif. Hence, by (H3), we have
f ( x , u α n ( x ) ) u α n ( x ) | u α n ( x ) | 2 | w α n ( x ) | 2 d x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_Equ11_HTML.gif
(3.8)
From (3.4), we obtain
1 ( 1 ) ( w 0 + w = 0 ) f ( x , u α n ( x ) ) u α n ( x ) | u α n ( x ) | 2 | w α n ( x ) | 2 d x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_Equ12_HTML.gif
(3.9)

By (3.8), the right-hand side of (3.9) + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq109_HTML.gif. This is a contradiction.

In any case, we obtain a contradiction. Therefore, { u α n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq110_HTML.gif is bounded.

Finally, we claim that for every m N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq54_HTML.gif,
I ( u ) as  u , u X m 1 X 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_Equao_HTML.gif
By (H2) and (H3), there exists large enough M such that
F ( x , t ) M t 2 C 6 , x Ω , t R . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_Equap_HTML.gif
So, for any u X m 1 X 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_IEq111_HTML.gif, we have
I ( t u ) = 1 2 t 2 Ω ( | u | 2 l | u | 2 ) d x Ω F ( x , t u ) d x 1 2 t 2 Ω ( | u | 2 l | u | 2 ) d x M t 2 Ω u 2 d x + C 6 | Ω | as  t + . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-154/MediaObjects/13661_2012_Article_252_Equaq_HTML.gif

Hence, our claim holds. □

Proof of Theorem 1.2 We omit the proof which depends on Lemma 2.2 and is similar to the preceding one. □

Author’s contributions

The author read and approved the final manuscript.

Declarations

Acknowledgements

The author would like to thank the referees for valuable comments and suggestions in improving this paper. This work was supported by the National NSF (Grant No. 10671156) of China.

Authors’ Affiliations

(1)
School of Mathematics and Statistics, Tianshui Normal University

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