Infinitely many solutions for quasilinear Schrödinger equations with sign-changing nonlinearity without the aid of 4-superlinear at infinity

1 Introduction

In this article, we are concerned with quasilinear Schrödinger equation of the form

where g ( x , u ) ∈ C ( R N × R , R ) is q -superlinear with 2 < q < min ( 4 , p ) , 2 < p < 2 2 ∗ 2 ∗ = 2 N N − 2 , N ≥ 3 , λ is positive parameter, and V ( x ) ∈ C ( R N , R + ) ∩ L ∞ ( R N ) .

We would like to mention that quasilinear Schrödinger equation (P) arises in various branches of mathematical physics and has been derived as models of several physics phenomenon see [1,2,3]. The quasilinear problem has been studied extensively in recent years with a huge variety of conditions on the potential V and the nonlinearity g , see, for example, [4,5, 6,7,8] and references therein.

Alves et al. [9] assumed nonlinearity to be 4-superlinear and that the potential V is continuous and satisfies:

( V 1 ) V ∈ C ( R N , R ) and V ( x ) ≥ V ( 0 ) > 0 , for all x ∈ R N .

( V 2 ) lim ∣ x ∣ → + ∞ V ( x ) = V ∞ and V ( x ) ≤ V ∞ , for all x ∈ R N .

Under these two conditions, they combined the variational method with perturbation arguments to obtain a solution. In [10,11], the authors obtained the multiplicity of solutions and made an assumption on V and also the nonlinearity to be 4-superlinear, where they have used variational methods. Most of the works requires the condition on the nonlinearity to be 4-superlinear in that sense p > 4 . This condition that makes the application of the Fountain theorem is fluid. Indeed, if 4 < p < 2 2 ∗ , the boundedness of the Cerami sequences of the energy functional associated with the problem (P) follows by compactness embedding of H 1 ( R N ) into L p 2 ( R N ) . So, the energy functional satisfies the Cerami condition for any c . Consequently, we obtain a set of unbounded of critical values. In contrast, if the nonlinear term g ( x , u ) changes its sign and without the aid of condition 4 < p < 2 2 ∗ , we will have difficulties to obtain the delamination of the Cerami sequences, as well as the convergence, to complete the hypotheses of the Fountain theorem [12]. To the best of our knowledge, there are only a few recent papers that deal with the nonlinearity to be not 4-superlinear and the sign-changing potential case for problem (P). Motivated by papers [10,11], in the present article, we shall consider problem (P) with more generally on the nonlinearity of g ( x , u ) . Namely, when powers of the nonlinearity are 2 < q < min ( 4 , p ) , 2 < p < 2 2 ∗ and can be allowed to sign-changing. By a dual approach and Fountain theorem, we establish the existence of infinitely many solutions. We state below our hypotheses on the potential V and on the nonlinearity g ( x , u ) in the subcritical case:

1. V ∈ C ( R N , R ) ∩ L ∞ ( R N ) and 0 < V 0 = inf x ∈ R N V ( x ) .

2. ( V ( x ) ) − 1 ∈ L 1 ( R N ) .

3. g ∈ C ( R N × R , R ) such that g ( x , − u ) = − g ( x , u ) for all x , u ∈ R N × R , there exit positive functions ξ ∈ L 4 q ′ ( R N ) ∩ L 2 ( R N ) , 4 q ′ is the conjugate of 4 q and τ ∈ L ∞ ( R N ) ∩ L 2 ( R N ) such that

   ∣ g ( x , t ) ∣ ≤ ξ ( x ) ∣ t ∣ q − 1 + τ ( x ) ∣ t ∣ p − 1 , ∀ ( x , t ) ∈ R N × R .

4. Let G ( x , t ) = ∫ 0 t g ( x , u ) d u , lim t → + ∞ ∣ G ( x , t ) ∣ t q = + ∞ uniformly in x , 2 < q < 4 and there exists r 0 > 0 such that G ( x , t ) ≥ 0 ∀ ( x , t ) ∈ R N × R , ∣ t ∣ ≥ r 0 .

5. There exists c 0 > 0 and δ 0 > 0 such that

   ∣ G ( x , t ) ∣ t 2 p p − 2 ≤ c 0 G ˜ ( x , t ) for all ( x , t ) ∈ R N × R \ [ − δ 0 , δ 0 ] ,

   where G ˜ ( x , t ) = 1 μ g ( x , t ) t − G ( x , t ) is the positive function and q < μ < p .

Notation. In this article, we make use of the following notation:

• C , C 0 , C 1 , C 2 , … denote positive (possibly different) constants.

• For 2 ≤ s < ∞ , the usual Lebesgue space is endowed with the norm

  ‖ u ‖ s = ‖ u ‖ L s ( R N ) ≔ ∫ R N ∣ u ( x ) ∣ s d x 1 / s , u ∈ L s ( R N ) .

• H 1 ( R N ) = { u ∈ L 2 ( R N ) : ∇ u ∈ L 2 ( R N ) } endowed with the norm

  ‖ u ‖ H 1 ( R N ) = ∫ R N ∣ ∇ u ( x ) ∣ 2 + u 2 d x 1 / 2 .

• S denotes the best constant that verifies

  (1) S = inf u ∈ H 1 ( R N ) \ { 0 } ∫ R N ∣ ∇ u ∣ 2 d x 1 / 2 ∫ R N ∣ u ∣ 2 ∗ d x 1 / 2 ∗ .

• For any s ∈ ( 2 , 2 ∗ ) , S s denote by

  (2) S s ≔ inf u ∈ H 1 ( R N ) \ { 0 } ‖ u ‖ H 1 ( R N ) ‖ u ‖ s .

2 Reformulation of the problem

From the embedding results, it is well known that H 1 ( R N ) ↪ L p ( R N ) is continuous for 2 ≤ p ≤ 2 ∗ and is compact when 2 ≤ p < 2 ∗ . For more general facts about the embedded, we refer the reader to the previous articles [13,14].

To find solutions of (P), we will use a variational approach. Hence, we will associate a suitable functional to our problem. More precisely, the Euler-Lagrange functional related to problem (P) is given by J λ : H 1 ( R N ) → R defined as follows:

Since,

we obtain

Because of the term ∫ R N ∣ u ∣ 2 ∣ ∇ u ∣ 2 d x , the functional J λ is not defined on space H 1 ( R N ) . For this, we follow the argument developed by Liu et al. [15] (see also [4]), and we change the variables v = f − 1 ( u ) , where f is defined by

Let us assemble some properties of the change of variables f : R → R , which will be used in the continued from the article.

Under this change the functional J λ will be

Obviously, I λ ∈ C 1 ( H 1 ( R N ) , R ) and

for any w ∈ H 1 ( R N ) .

Moreover, the critical points of I λ are the weak solutions of the following equation:

Considering that if v is a critical point of the functional I λ , so u = f ( v ) is a critical point of the functional J λ , that is, u = f ( v ) is a solution of problem (P).

3 The boundedness of the Cerami sequences

The first step for the ( C ) c -condition to hold is bounded.