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Multiple solutions of semilinear elliptic systems on the Heisenberg group

Boundary Value Problems20132013:157

DOI: 10.1186/1687-2770-2013-157

Received: 17 November 2012

Accepted: 10 June 2013

Published: 1 July 2013

Abstract

In this paper, a class of semilinear elliptic systems which have a strong resonance at the first eigenvalue on the Heisenberg group is considered. Under certain assumptions, by virtue of the variational methods, the multiple weak solutions of the systems are obtained.

MSC:35J20, 35J25, 65J67.

Keywords

semilinear elliptic system strong resonance variational method Heisenberg group

1 Introduction

Let H N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq1_HTML.gif be the space R N × R N × R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq2_HTML.gif equipped with the following group operation:
η η = ( x , y , t ) ( x , y , t ) = ( x + x , y + y , t + t + 2 ( x y x y ) ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_Equa_HTML.gif
where ‘’ denotes the usual inner-product in R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq3_HTML.gif. This operation endows H N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq1_HTML.gif with the structure of a Lie group. The vector fields X 1 , , X N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq4_HTML.gif, Y 1 , , Y N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq5_HTML.gif, T, given by
X j = x j + 2 y j t , Y j = y j 2 x j t , T = t , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_Equb_HTML.gif

form a basis for the tangent space at η = ( x , y , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq6_HTML.gif.

Definition 1.1 The Heisenberg Laplacian is by definition
Δ H = j = 1 N ( X j 2 + Y j 2 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_Equc_HTML.gif

and let H u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq7_HTML.gif denote the 2N-vector ( X 1 u , , X N u , Y 1 u , , Y N u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq8_HTML.gif.

Definition 1.2 The space S 0 1 , 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq9_HTML.gif is defined as the completion of C 0 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq10_HTML.gif in the norm
u S 0 1 , 2 2 = Ω j = 1 N ( | X j u | 2 + | Y j u | 2 ) = Ω | H u | 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_Equd_HTML.gif

Some existence and nonexistence for the semilinear equations or systems on the Heisenberg group have been studied by Garofalo, Lanconelli and Niu, see [1, 2], etc.

In this paper, we study the problems on the existence and multiplicity of solutions for the system
{ Δ H ( u v ) = λ 1 ( a ( x ) b ( x ) b ( x ) d ( x ) ) ( u v ) ( f ( x , u , v ) g ( x , u , v ) ) , x Ω , u = v = 0 , x Ω , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_Equ1_HTML.gif
(1.1)

where Ω H N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq11_HTML.gif is a bounded smooth domain, a , b , d C 0 ( Ω ¯ , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq12_HTML.gif and f , g C 1 ( Ω ¯ × R 2 , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq13_HTML.gif. Moreover, we assume that there is some function F ( x , u , v ) C 2 ( Ω ¯ × R 2 , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq14_HTML.gif such that F = ( f g ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq15_HTML.gif. Here F denotes the gradient in the variable u and v, i.e., F u = f https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq16_HTML.gif, F v = g https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq17_HTML.gif.

In fact, the condition in R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq3_HTML.gif was studied by da Silva; we can see [3]. In this paper we study the problem on the Heisenberg group H N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq1_HTML.gif. The elliptic problems at resonance have been studied by many authors; see [47].

We use the variation methods to solve problem (1.1). Finding weak solutions of (1.1) in E = S 0 1 , 2 ( Ω ) × S 0 1 , 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq18_HTML.gif is equivalent to finding critical points of the C 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq19_HTML.gif functional given by
I ( h ) = 1 2 h 2 1 2 Ω A h , h + Ω F ( x , h ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_Equ2_HTML.gif
(1.2)
where
h E , h = ( h ( 1 ) h ( 2 ) ) , h 2 = Ω | H h ( 1 ) | 2 + | H h ( 2 ) | 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_Eque_HTML.gif

and , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq20_HTML.gif denotes the usual inner product in R 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq21_HTML.gif.

We introduce the eigenvalue problem with weights. Let us denote by A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq22_HTML.gif the set of all continuous, cooperative and symmetric matrices A of order 2, given by
A ( x ) = ( a ( x ) b ( x ) b ( x ) d ( x ) ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_Equf_HTML.gif

where the functions a , b , d C ( Ω ¯ , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq23_HTML.gif satisfy the following conditions:

(A1) A ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq24_HTML.gif is cooperative, that is, b ( x ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq25_HTML.gif.

(A2) There is an x 0 Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq26_HTML.gif such that a ( x 0 ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq27_HTML.gif or d ( x 0 ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq28_HTML.gif.

Given A A ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq29_HTML.gif, consider the weighted eigenvalue problem
{ Δ H ( h ( 1 ) h ( 2 ) ) = λ A ( x ) ( h ( 1 ) h ( 2 ) ) , in  Ω , h ( 1 ) = h ( 2 ) = 0 , on  Ω , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_Equg_HTML.gif
if A A ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq29_HTML.gif. By virtue of the spectral theory for compact operators, we obtain the sequence of eigenvalues
0 < λ 1 < λ 2 λ 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_Equh_HTML.gif
such that λ k + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq30_HTML.gif as k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq31_HTML.gif; see [6, 8, 9]. Here, each eigenvalue λ k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq32_HTML.gif, k 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq33_HTML.gif has finite multiplicity, and we have
1 λ k = sup { Ω A h , h , h = 1 , h V k 1 } , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_Equi_HTML.gif

where V k = span { Φ 1 , , Φ k } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq34_HTML.gif with k 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq33_HTML.gif.

Remark 1.1
  1. (1)

    E = V k V k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq35_HTML.gif for k 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq33_HTML.gif.

     
  2. (2)
    The following variational inequalities hold:
    h 2 λ k Ω A h , h , h V k , k 1 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_Equ3_HTML.gif
    (1.3)
    h 2 λ k + 1 Ω A h , h , h V k , k 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_Equ4_HTML.gif
    (1.4)
     

The variational inequalities will be used in the next section. We would like to mention that the Φ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq36_HTML.gif is positive in Ω. In the paper, without loss of generality, we assume that λ 1 = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq37_HTML.gif.

We now state the assumptions and the main results in this paper. Firstly, we define the following functions:
{ T + = lim inf ( u , v ) ( , ) F ( x , u , v ) , S + = lim sup ( u , v ) ( , ) F ( x , u , v ) , T = lim inf ( u , v ) ( , ) F ( x , u , v ) , S = lim sup ( u , v ) ( , ) F ( x , u , v ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_Equ5_HTML.gif
(1.5)

The above functions belong to L 1 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq38_HTML.gif and the limits are taken a.e. and uniformly in x Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq39_HTML.gif.

Now we make the following basic hypotheses:

(E0) There exists k C ( Ω ¯ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq40_HTML.gif such that
lim | h | F ( x , h ) = 0 , | F ( x , h ) | k ( x ) , a.e.  x Ω , h R 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_Equj_HTML.gif

(E1) F ( x , h ) 1 2 ( 1 λ 2 ) A h , h + b 1 | Ω | 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq41_HTML.gif, b 1 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq42_HTML.gif, ( x , h ) Ω × R 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq43_HTML.gif.

(E2) A h , h 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq44_HTML.gif, ( x , h ) Ω × R 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq43_HTML.gif.

(E3) There exist α ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq45_HTML.gif and δ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq46_HTML.gif such that
F ( x , h ) 1 α 2 A h , h , x Ω  and  | z | < δ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_Equk_HTML.gif

(E4) Ω S + 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq47_HTML.gif and Ω S 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq48_HTML.gif.

(E5) There exists t 0 R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq49_HTML.gif such that
Ω F ( x , t 0 Φ 1 ) < min { Ω T + , Ω T } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_Equl_HTML.gif
(E6) There are t 1 < 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq50_HTML.gif and t 1 + > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq51_HTML.gif such that
Ω F ( x , t 1 ± Φ 1 ) < min { Ω T + , Ω T } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_Equm_HTML.gif

We can prove that the associated functional J has the saddle geometry. Actually, we have the following results.

Theorem 1.1 Let Ω H N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq11_HTML.gif be a bounded smooth domain, a ( x ) , b ( x ) , d ( x ) C 0 ( Ω ¯ , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq52_HTML.gif and f ( x , u , v ) , g ( x , u , v ) C 1 ( Ω ¯ × R 2 , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq53_HTML.gif. Assume that there is some function F ( x , u , v ) C 2 ( Ω ¯ × R 2 , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq54_HTML.gif such that F u = f https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq16_HTML.gif, F v = g https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq55_HTML.gif. Furthermore, if the conditions (E0), (E1), (E2) are satisfied, problem (1.1) has at least one solution z 1 E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq56_HTML.gif.

Remark 1.2 For the hypotheses F ( x , 0 , 0 ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq57_HTML.gif and F ( x , 0 , 0 ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq58_HTML.gif, problem (1.1) admits the trivial solution ( u , v ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq59_HTML.gif. In this case, the main point is to assure the existence of nontrivial solutions.

Theorem 1.2 Let Ω H N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq11_HTML.gif be a bounded smooth domain, a ( x ) , b ( x ) , d ( x ) C 0 ( Ω ¯ , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq52_HTML.gif and f ( x , u , v ) , g ( x , u , v ) C 1 ( Ω ¯ × R 2 , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq53_HTML.gif. Assume that there is some function F ( x , u , v ) C 2 ( Ω ¯ × R 2 , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq54_HTML.gif such that F u = f https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq16_HTML.gif, F v = g https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq55_HTML.gif. Furthermore, if the conditions (E0), (E2), (E3), (E4) and (E5) are satisfied, then problem (1.1) has at least two nontrivial solutions.

Theorem 1.3 Let Ω H N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq11_HTML.gif be a bounded smooth domain, a ( x ) , b ( x ) , d ( x ) C 0 ( Ω ¯ , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq52_HTML.gif and f ( x , u , v ) , g ( x , u , v ) C 1 ( Ω ¯ × R 2 , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq60_HTML.gif. Assume that there is some function F ( x , u , v ) C 2 ( Ω ¯ × R 2 , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq54_HTML.gif such that F u = f https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq16_HTML.gif, F v = g https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq55_HTML.gif. Furthermore, if the conditions (E0), (E1), (E2), (E3), (E4) and (E6) are satisfied, then problem (1.1) has at least three nontrivial solutions.

2 Preliminaries and fundamental lemmas

In this section, we prove some lemmas needed in the proof of our main theorems.

We first introduce the Folland-Stein embedding theorem (see [10]) as follows.

Lemma 2.1 Let Ω H N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq11_HTML.gif be a bounded domain and let Q = 2 N + 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq61_HTML.gif. Then S 0 1 , 2 ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq9_HTML.gif compactly embedding in L p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq62_HTML.gif, where 2 p < 2 Q Q 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq63_HTML.gif.

To establish Lemmas 2.7 and 2.8, we introduce the following corollary of the Ekeland variation principle (see [11]).

Lemma 2.2 X is a metric space, I C 1 ( X , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq64_HTML.gif is bounded from below, which satisfies the ( P S ) c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq65_HTML.gif condition, then c = inf x X E ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq66_HTML.gif is a critical value of E.

Next, we describe some results under the geometry for the functional I.

Lemma 2.3 Under hypotheses (E0) and (E1), the functional I has the following saddle geometry:

(L3-1) I ( h ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq67_HTML.gif if h https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq68_HTML.gif with h V 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq69_HTML.gif.

(L3-2) There is α R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq70_HTML.gif such that I ( h ) α https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq71_HTML.gif, z V 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq72_HTML.gif.

(L3-3) I ( h ) b 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq73_HTML.gif, z V 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq74_HTML.gif.

Proof (L3-1). From (1.2), (1.4) we have
I ( h ) 1 2 ( 1 1 λ 2 ) h 2 + Ω F ( x , h ) , h V 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_Equn_HTML.gif

Using (E0), we have J ( h ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq75_HTML.gif, as h https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq68_HTML.gif.

(L3-2). By simple calculation, we get
I ( h ) = Ω F ( x , h ) , h V 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_Equo_HTML.gif
By using (E0), we have
I ( h ) = Ω F ( x , h ) Ω k ( x ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_Equp_HTML.gif

So, we choose α = Ω k ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq76_HTML.gif.

(L3-3). By (E1) and the variational inequality (1.4), we have
I ( h ) = 1 2 h 2 1 2 Ω A h , h + Ω F ( x , h ) 1 2 h 2 λ 2 2 Ω A h , h + b 1 b 1 , z V 1 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_Equq_HTML.gif

the proof of this lemma is completed. □

Next, we prove the Palais-Smale conditions at some levels for the functional I. We recall that I: E R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq77_HTML.gif is said to satisfy the Palais-Smale conditions at the level c R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq78_HTML.gif ( ( P S ) c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq65_HTML.gif in short) if any sequence { h n } E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq79_HTML.gif such that
I ( h n ) c , I ( h n ) 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_Equr_HTML.gif

as n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq80_HTML.gif, possesses a convergent subsequence in E. Moreover, we say that I satisfies the (PS) conditions when we have ( P S ) c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq65_HTML.gif for all c R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq78_HTML.gif.

Lemma 2.4 Assume that the condition (E0) holds. Then the functional I has the ( P S ) c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq65_HTML.gif conditions whenever c < min { Ω T + , Ω T } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq81_HTML.gif or c > max { Ω S + , Ω S } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq82_HTML.gif.

Proof We only prove the condition for all c < min { Ω T + , Ω T } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq83_HTML.gif. For the case c > max { Ω S + , Ω S } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq84_HTML.gif, we can use similar methods.
  1. 1.

    Boundedness of the (PS) sequence.

     

The proof is by contradiction. Suppose that there exists a ( P S ) c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq65_HTML.gif unbounded sequence { h n } E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq85_HTML.gif such that c < min { Ω T + , Ω T } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq86_HTML.gif. For the ease of notation and without loss of generality, we assume that

h n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq87_HTML.gif,

I ( h n ) c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq88_HTML.gif,

I ( h n ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq89_HTML.gif, n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq80_HTML.gif.

We define h ¯ n = h n h n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq90_HTML.gif, hence there is an h ¯ E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq91_HTML.gif with the following properties:

h ¯ n h ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq92_HTML.gif in E,

h ¯ n h ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq93_HTML.gif in L p ( Ω ) × L p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq94_HTML.gif, where 2 p < 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq95_HTML.gif and 2 = 2 N + 2 N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq96_HTML.gif,

h ¯ n h ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq93_HTML.gif a.e. in Ω.

For any Φ E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq97_HTML.gif, obviously I ( h n ) Φ h n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq98_HTML.gif. By simple calculation, it is easy to obtain
I ( h n ) Φ = Ω H h n , H Φ Ω A h n , Φ + Ω F ( x , h n ) , Φ , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_Equs_HTML.gif
where h n = ( h n ( 1 ) h n ( 2 ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq99_HTML.gif, Φ = ( Φ ( 1 ) Φ ( 2 ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq100_HTML.gif. We have
I ( h n ) Φ h n = Ω H h ¯ n , H Φ Ω A h ¯ n , Φ + Ω F ( x , h n ) , Φ h n 0 as  n . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_Equt_HTML.gif
From the convergence of { h ¯ n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq101_HTML.gif, we have
Ω H h ¯ , H Φ = Ω A h ¯ , Φ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_Equu_HTML.gif

We see that λ 1 = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq37_HTML.gif, and by the definition of λ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq102_HTML.gif, we obtain that h ¯ = ± Φ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq103_HTML.gif. So, we suppose initially that h ¯ = Φ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq104_HTML.gif. Because Φ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq36_HTML.gif is positive, i.e., Φ 1 ( 1 ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq105_HTML.gif, Φ 1 ( 2 ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq106_HTML.gif, it is obvious that h n ( 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq107_HTML.gif, h n ( 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq108_HTML.gif, x Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq109_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq80_HTML.gif.

Hence, we can take h n = t n Φ 1 + ω n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq110_HTML.gif, where { t n } R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq111_HTML.gif, { ω n } V 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq112_HTML.gif, and we have
I ( h n ) = 1 2 t n Φ 1 + ω n 2 1 2 Ω A ( t n Φ 1 + ω n ) , t n Φ 1 + ω n + Ω F ( x , h n ) = 1 2 ω n 2 Ω A ω n , ω n + Ω F ( x , h n ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_Equv_HTML.gif
Using (1.4), we obtain
I ( h n ) 1 2 ( 1 1 λ 2 ) ω n 2 + Ω F ( x , h n ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_Equ6_HTML.gif
(2.1)

Since I ( h n ) c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq88_HTML.gif, it is easy to obtain that the sequence { ω n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq113_HTML.gif is bounded. On the other hand, because of h n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq87_HTML.gif, on a subsequence | t n | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq114_HTML.gif, without loss of generality, we assume t n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq115_HTML.gif.

Now, using Hölder’s inequality and (E0), we have
| Ω F ( x , h n ) ω n | C ( Ω | F ( x , h n ) | 2 ) 1 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_Equw_HTML.gif
Thus, applying the dominated convergence theorem, we conclude that
lim n Ω F ( x , h n ) ω n = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_Equ7_HTML.gif
(2.2)
On the other hand,
I ( h n ) ω n = Ω H h n , H ω n Ω A h n , ω n + Ω F ( x , h n ) , ω n = ω n 2 Ω A h n , ω n + Ω F ( x , h n ) , ω n . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_Equx_HTML.gif
Using (2.2), (1.4), we obtain
( 1 1 λ 2 ) ω n 2 | I ( h n ) ω n | + | Ω F ( x , h n ) , ω n | 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_Equy_HTML.gif
as n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq80_HTML.gif. Therefore, by variational inequalities (1.3) and (1.4), we obtain that
ω n 2 Ω A h n , ω n 0 as  n . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_Equz_HTML.gif
Consequently, by virtue of Fatou’s lemma and (E0), we have
c = lim n ( 1 2 ω n 2 Ω A ω n , ω n + Ω F ( x , h n ) ) = lim inf n Ω F ( x , t n Φ 1 + ω n ) Ω lim inf n F ( x , t n Φ 1 + ω n ) = Ω T , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_Equaa_HTML.gif
which contradicts the condition c < min { Ω T + , Ω T } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq116_HTML.gif. Hence, the ( P S ) c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq65_HTML.gif sequence is bounded.
  1. 2.

    Various convergence of { h n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq117_HTML.gif.

     

Since { h n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq117_HTML.gif is a bounded sequence, there is an h E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq118_HTML.gif with the following properties:

h n h https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq119_HTML.gif in E,

h n h https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq120_HTML.gif in L p ( Ω ) × L p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq121_HTML.gif, where 2 p < 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq95_HTML.gif and 2 = 2 N + 2 N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq96_HTML.gif,

h n h https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq120_HTML.gif a.e. in Ω.

  1. 3.

    { h n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq117_HTML.gif convergence to h in E.

     
From the definition of ( P S ) c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq65_HTML.gif sequence, we have, as n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq122_HTML.gif,
I ( h n ) h = Ω H h n , H h Ω A h n , h + Ω F ( x , h n ) , h 0 , I ( h n ) h n = Ω | H h n | 2 Ω A h n , h n + Ω F ( x , h n ) , h n 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_Equab_HTML.gif
By Fatou’s lemma and the above convergence of { h n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq117_HTML.gif, it is easy to show that
Ω A h n , h Ω A h , h , Ω F ( x , h n ) , h Ω F ( x , h ) , h , Ω A h n , h n Ω A h , h , Ω F ( x , h n ) , h n Ω F ( x , h ) , h https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_Equac_HTML.gif
as n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq80_HTML.gif. Hence, we have
Ω H h n , H h Ω A h , h Ω F ( x , h ) , h as  n , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_Equ8_HTML.gif
(2.3)
Ω | H h n | 2 Ω A h , h Ω F ( x , h ) , h as  n . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_Equ9_HTML.gif
(2.4)
By weak convergence, we have
Ω H h n , H h Ω H h , H h as  n . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_Equ10_HTML.gif
(2.5)
Using (2.3), (2.4) and (2.5), by simple calculation, we obtain
Ω | H h n H h | 2 0 as  n . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_Equad_HTML.gif

The proof is completed. □

Lemma 2.5 Suppose that (E0) and (E3) are satisfied. Then the origin is a local minimum for the functional I.

Proof Using (E3), we can choose p ( 2 , 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq123_HTML.gif and a constant C > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq124_HTML.gif such that
F ( x , h ) 1 α 2 A h , h C | h | p , ( x , h ) Ω × R 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_Equae_HTML.gif
Consequently, we have
I ( h ) = 1 2 h 2 1 2 Ω A h , h + Ω F ( x , h ) 1 2 ( 1 α ) h 2 C Ω | h | p 1 2 ( 1 α ) h 2 C h p 1 4 ( 1 α ) h 2 , h < ρ , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_Equaf_HTML.gif

where ρ is small enough and 0 < ρ < t 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq125_HTML.gif, t 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq126_HTML.gif is provided by (E5). Therefore the proof has been completed. □

To complete the mountain pass geometry, we prove the following result.

Lemma 2.6 Let the hypotheses (E0), (E4) and (E5) hold. Then there exist h 0 E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq127_HTML.gif and ρ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq128_HTML.gif such that I ( h 0 ) < 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq129_HTML.gif and h 0 > ρ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq130_HTML.gif.

Proof Using (E2) and (E5), we take h 0 = t 0 Φ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq131_HTML.gif, where t 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq126_HTML.gif is provided by (E5). Thus, we obtain
I ( t 0 Φ 1 ) = 1 2 t 0 Φ 1 2 1 2 Ω A ( t 0 Φ 1 ) , t 0 Φ 1 + Ω F ( x , t 0 Φ 1 ) , = Ω F ( x , t 0 Φ 1 ) < min { Ω T + , Ω T } < max { Ω S + , Ω S } 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_Equag_HTML.gif

and t 0 Φ 1 = t 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq132_HTML.gif. If we take 0 < ρ < t 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq133_HTML.gif, then the conclusion follows. □

Lemma 2.7 Under hypotheses (E0), (E4) and (E5), problem (1.1) has at least one nontrivial solution h 0 E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq127_HTML.gif. Moreover, h 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq134_HTML.gif has negative energy, i.e., J ( h 0 ) < 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq135_HTML.gif.

Proof By (E0) and (1.4), we obtain
I ( h ) = 1 2 h 2 1 2 Ω A h , h + Ω F ( x , h ) Ω F ( x , h ) Ω k ( x ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_Equah_HTML.gif
Therefore, the functional I is bounded below. In this case, we would like to mention that I has the ( P S ) c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq65_HTML.gif conditions with c = inf { I ( h ) : h E } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq136_HTML.gif. For seeing this, by Lemma 2.4, we take t 0 R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq137_HTML.gif provided by (E5) we can obtain
c I ( t Φ 1 ) = Ω F ( x , t Φ 1 ) < min { Ω T + , Ω T } 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_Equai_HTML.gif

Consequently, applying Lemma 2.2, we have one critical point h 0 E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq127_HTML.gif such that I ( h 0 ) = inf { I ( h ) : h E } I ( t Φ 1 ) < 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq138_HTML.gif. The proof of this lemma is completed. □

To prove Theorem 1.3, we establish the following lemma.

Lemma 2.8 Assume that the conditions (E0), (E1), (E4) and (E6) hold. Then problem (1.1) has at least two nontrivial solutions with negative energy.

Proof Define
M + = { t Φ 1 + ω , t 0 , ω V 1 } , M = { t Φ 1 + ω , t 0 , ω V 1 } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_Equaj_HTML.gif

We have M + = M = V 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq139_HTML.gif. Hence, we minimize the functional I restricted to M + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq140_HTML.gif and M https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq141_HTML.gif.

Firstly, we consider the functionals I ± = I | M ± https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq142_HTML.gif. Using Lemma 2.4, I ± https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq143_HTML.gif possesses the ( P S ) c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq65_HTML.gif conditions whenever c < min { Ω T + , Ω T } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq116_HTML.gif. Therefore, we obtain that I ± https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq143_HTML.gif satisfies the ( P S ) c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq65_HTML.gif conditions with c ± = inf { I ± ( h ) : h M ± } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq144_HTML.gif.

In this way, by using Lemma 2.2 for the functional I ± https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq143_HTML.gif, we obtain two critical points which we denote by h 0 + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq145_HTML.gif and h 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq146_HTML.gif, respectively. Thus, we have c + = I + ( h 0 + ) = inf h M + { I ( h ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq147_HTML.gif and c = I ( h 0 ) = inf h M { I ( h ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq148_HTML.gif.

Moreover, we affirm that h 0 + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq145_HTML.gif and h 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq146_HTML.gif are nonzero critical points. To see this, from (E4) and (E6), we obtain that
I ± ( h 0 ± ) I ± ( t 1 ± Φ 1 ) = Ω F ( x , t 1 ± Φ 1 ) < min { Ω T + , Ω T } 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_Equak_HTML.gif
and I restricted to V 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq149_HTML.gif is nonnegative. More specifically, given ω V 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq150_HTML.gif, using (L3-3) in Lemma 2.3, we have
I ( ω ) b 1 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_Equ11_HTML.gif
(2.6)

Next, we prove that h 0 + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq145_HTML.gif and h 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq146_HTML.gif are distinct. The proof of this affirmation is by contradiction. If h 0 + = h 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq151_HTML.gif, then h 0 + = h 0 V 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq152_HTML.gif. Using (2.6), we obtain I ( h 0 + ) < 0 I ( h 0 + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq153_HTML.gif. Therefore, we have a contradiction. Consequently, we get h 0 + h 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq154_HTML.gif. Thus problem (1.1) has at least two nontrivial solutions. Moreover, these solutions have negative energy. □

3 Proof of main theorems

In this section, we prove Theorem 1.1, Theorem 1.2 and Theorem 1.3.

Proof of Theorem 1.1 From Lemma 2.4, the functional I satisfies the ( P S ) c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq65_HTML.gif conditions for some levels c R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq78_HTML.gif. Set E = V 1 V 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq155_HTML.gif, where V 1 = span { Φ 1 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq156_HTML.gif. Using Lemma 2.3, we get that the functional I satisfies the saddle point geometry (see [12], Theorem 1.11). This implies that I has one critical point h 1 E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq157_HTML.gif. Theorem 1.1 is proved. □

Proof of Theorem 1.2 From Lemma 2.5 and Lemma 2.6, we know that the functional I satisfies the geometric conditions of the mountain pass theorem. Moreover, the functional I satisfies the ( P S ) c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq65_HTML.gif conditions for all c 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq158_HTML.gif. Thus, we have a solution h 2 E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq159_HTML.gif given by the mountain pass theorem. Obviously, the solution h 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq160_HTML.gif satisfies I ( h 2 ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq161_HTML.gif.

On the other hand, by Lemma 2.7, we get another solution h 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq134_HTML.gif and I ( h 0 ) < 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq129_HTML.gif. It follows that problem (1.1) has at least two nontrivial solutions. The proof is completed. □

Proof of Theorem 1.3 Since the conditions (E0), (E3), (E4) and (E5) imply that Lemma 2.5 and Lemma 2.6 hold. Thus, we have one solution h 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq160_HTML.gif which satisfies I ( h 2 ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq161_HTML.gif.

On the other hand, using Lemma 2.8, we obtain two distinct critical points h 0 ± https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq162_HTML.gif such that I ( h 0 ± ) < 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-157/MediaObjects/13661_2012_Article_411_IEq163_HTML.gif. Therefore, we obtain that problem (1.1) has at least three nontrivial solutions. The proof is completed. □

Declarations

Acknowledgements

The authors express their sincere thanks to the referees for their valuable criticism of the manuscript and for helpful suggestions. This work was supported by the National Natural Science Foundation of China (11171220) and Shanghai Leading Academic Discipline Project (XTKX2012).

Authors’ Affiliations

(1)
College of Science, University of Shanghai for Science and Technology

References

  1. Garofalo N, Lanconelli E: Existence and nonexistence results for semilinear equations on the Heisenberg group. Indiana Univ. Math. J. 1992, 41: 71–98. 10.1512/iumj.1992.41.41005MathSciNetView Article
  2. Niu PC: Nonexistence for semilinear equations and systems in the Heisenberg group. J. Math. Anal. Appl. 1999, 240: 47–59. 10.1006/jmaa.1999.6574MathSciNetView Article
  3. Silva, DA: Multiplicity of solutions for gradient systems under strong resonance at a the first eigenvalue. arXiv:1206.7097v1. http://​www.​e-printatarXiv.​org
  4. Ahmad S, Lazer AC, Paul JL: Elementary critical point theory and perturbations of elliptic boundary value problems at resonance. Indiana Univ. Math. J. 1976, 25: 933–944. 10.1512/iumj.1976.25.25074MathSciNetView Article
  5. Bartsch T, Li SJ: Critical point theory for asymptotically quadratic functionals and applications to problems with resonance. Nonlinear Anal. TMA 1997, 28: 419–441. 10.1016/0362-546X(95)00167-TMathSciNetView Article
  6. Furtado FE, De Paiva FO: Multiplicity of solutions for resonant elliptic systems. J. Math. Anal. Appl. 2006, 319: 435–449. 10.1016/j.jmaa.2005.06.038MathSciNetView Article
  7. Landesman EM, Lazer AC: Nonlinear perturbations of linear elliptic boundary value problems at resonance. J. Math. Mech. 1969/1970, 19: 609–623.MathSciNet
  8. Chang KC: Principal eigenvalue for weight in elliptic systems. Nonlinear Anal. 2001, 46: 419–433. 10.1016/S0362-546X(00)00140-1MathSciNetView Article
  9. De Figueiredo DG: Positive solutions of semilinear elliptic problems. Lecture Note in Math. 957. In Differential Equations. Springer, Berlin; 1982.View Article
  10. Sara M: Infinitely many solutions of a semilinear problem for the Heisenberg Laplacian on the Hersenberg group. Manuscr. Math. 2005, 116: 357–384. 10.1007/s00229-004-0534-1View Article
  11. Willem M: Minimax Theorems. Progress in Nonlinear Differential Equations and Their Applications. Birkhäuser, Basel; 1996.
  12. Silva EA: Linking theorems and applications to semilinear elliptic problems at resonance. Nonlinear Anal. 1991, 16: 455–477. 10.1016/0362-546X(91)90070-HMathSciNetView Article

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© Jia et al.; licensee Springer. 2013

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