INFINITELY MANY SOLUTIONS FOR GENERALIZED QUASILINEAR SCHR¨ODINGER EQUATIONS WITH SIGN-CHANGING POTENTIAL

. We investigate a class of generalized quasilinear Schr¨odinger equations where g ( u ) : R → R + is a nondecreasing function with respect to | u | , the potential V ( x ) and the primitive of the nonlinearity f ( x,u ) are allowed to be sign-changing. Under some suitable assumptions, we obtain the existence of inﬁnitely many nontrivial solutions. The proof is based on a change of variable as well as symmetric Mountain Pass Theorem.

1. Introduction and main results. We are concerned with the following generalized quasilinear Schrödinger equations: −div(g 2 (u)∇u) + g(u)g (u)|∇u| 2 + V (x)u = f (x, u) in R N , (1.1) where N ≥ 1 and f ∈ C(R N × R, R). Moreover, we assume the function g(u) and the potential V (x) satisfy the following assumptions: (g) g(u) : R → R + is an even function with g (u) ≥ 0 for all u ≥ 0, ug (u) < g(u) for all u ∈ R and g (u) ≥ 0 is strict on a subset of positive measure in R; (V 1 ) V ∈ C(R N , R) and inf x∈R N V (x) > −∞; (V 2 ) for each M > 0, there exists a constant r > 0 such that where meas(·) denotes the Lebesgue measure in R N .
The quasilinear Schrödinger equations, of the form similar to problem (1.1), arise in various branches of mathematical physics, and they are related to the existence of standing wave solutions for quasilinear Schrödinger equations where z : R × R N → C, W : R N → R is a given potential, l and k(x, z) are real functions. We would like to mention that quasilinear equations of the form (1.2) have been derived as models of several physical phenomena corresponding to various types on nonlinear terms l. For instance, in the case of l(s) = s, problem (1.2) was used as a model of the time evolution of the condensate wave function in superfluid film, and was called the superfluid film equation in fluid mechnics by Kurihura (see [10]); the case l(s) = (1 + s) 1 2 models the self-channeling of a high-power ultrashort laser in matter, see [2,3,4,19]. Equation (1.2) also appears in plasma physics and fluid mechanics [10,11,14,17], in mechanics [9] and in condensed matter theory [15].
Putting z(t, x) = exp(−iEt)u(x) in (1.2), where E ∈ R and u(x) > 0 is a real function, we obtain a corresponding of elliptic type − u + V (x)u − (l(|u| 2 ))l (|u| 2 )u = k(x, u), x ∈ R N . (1.3) If we let g 2 (u) = 1 + [(l(u 2 )) ] 2 2 , then (1.3) turns into (1.1) (see [8,20]). And if we set g 2 (u) = 1 + 2u 2 , i.e., l(s) = s, we get the superfluid film equation in plasma physics: In the last decades, the existence and multiplicity of nontrivial solutions for quasilinear problem (1.4) have begun to receive much attention. We refer the readers to [12,13,16,24,25,27,28,29] and the reference therein. It is worth pointing out that, among these works, the method of the change of variable introduced by Colin and Jeanjean [5] and Liu, Wang and Wang [13] was proved to be a powerful tool to investigate the existence and multiplicity of solutions for problem (1.4).
In the past, the research on the existence of standing wave solutions for equation (1.3) mainly concentrates upon some given special function l(s), a natural question is whether there is a unified method to study (1.3) with general functions l(s)?
Recently, Shen and Wang [20] have given an affirmative answer and obtained the existence of positive solutions for (1.1) when f (x, u) is superlinear and subcritical. We must point out that the authors introduced a new change of variable in this work which provides us a useful method in studying problem (1.1). Later, the results was extended by Deng et al. in [7,8], and they established the existence of positive solutions when f (x, u) is critical. In [6], Deng et al. constructed the existence of infinitely many sign-changing solutions for problem (1.1) under some assumptions on g, f and V . And we have also studied the existence and multiplicity of solutions for problem (1.1) by using this change of variable in [21,22,23].
However, in the aforementioned papers [6,7,8,20,21,22,23], the authors all assumed the potential V (x) is non-negative. As far as we know there are no papers dealing with the case which the problem (1.1) has a sign-changing potential. We also note that in most of the above papers, the condition of the type of Ambrosetti-Rabinowitz (or shortly, (AR)-condition), that is (f ) there exists 2 < µ < 2 * such that for any u = 0, there holds 0 < µg(u) is necessary. However, there are many functions which do not satisfy the condition (f ).
Motivated by all facts mentioned above, it is very natural for us to pose some questions as follows: (Q 1 ) Can one establish the suitable variational framework for problem (1.1) with the sign-changing potential, precisely, the potential V (x) satisfies the assumptions (V 1 ) and (V 2 ). (Q 2 ) If the condition (f ) is replaced by a weaker condition or some other suitable conditions in problem (1.1), will the problem admit one or multiple nontrivial solutions?
In the present paper, we restrict our attention to the existence of infinitely many solutions for problem (1.1) and try to seek definite answers to questions (Q 1 ) and (Q 2 ).
Before stating our main results, we need to introduce some notations.
Notation. Throughout this paper, we denote 2 * = ∞ if N = 1, 2 and 2 * = 2N N −2 if N ≥ 3; C and C i will denote various positive constants; the strong (respectively weak) convergence is denoted by → (respectively ); o(1) denotes any quantity which tends to zero when n → ∞; B ρ (0) denotes a ball centered at the origin with radius ρ > 0. Now we make the following assumptions: (f 1 ) f ∈ C(R N × R, R), and there exist constants c 1 , c 2 > 0 such that for all u ∈ R, |G(u)| 2 = ∞ uniformly in x and there exists r 0 > 0 such that F (x, u) ≥ 0 for all x ∈ R N and |u| ≥ r 0 ; − F (x, u) ≥ 0, and there exist c 3 > 0 and σ > max{1, N 2 } such that We summarize our main results as follows: are satisfied, then problem (1.1) possesses infinitely many solutions {u n } such that u n → ∞ and I(u n ) → ∞ (I will be defined later). Remark 1. From (V 1 ) and (f 2 ), we easily see that V (x) and F (x, u) are allowed to be sign-changing. Moreover, the usual "superlinear condition" at the origin f (x, u) = o(u) uniformly in x as u → 0, which is all assumed in the aforementioned works for problem (1.1), is not needed in our result.
Remark 2. Note that g is an even function with g (u) ≥ 0 for all u ≥ 0, and ug (u) < g(u) for all u ∈ R, it is not difficult to deduce that |g(u)| ≤ c 4 |u| + c 5 and g (u) ≤ c 6 , with some constants c 4 , c 5 , c 6 > 0.
Before proceeding to the proof of the main results, we give an example to illustrate the above assumptions.
, by a simple computation, it is easy to check that the function satisfies the assumptions (f 1 )−(f 4 ), but does not satisfy condition (f ).
The rest of the paper is organized as follows. After presenting some preliminary results in section 2, we give the proof of our main result in section 3.

2.
Variational setting and preliminaries. Before establishing the variational framework for problem (1.1), we first notice a fact: from (V 1 ), we easily see that We note that problem (2.1) is equivalent to the problem (1.1) and the hypotheses ( In what follows, we just need to study the equivalent problem (2.1). Throughout this section, we assume that g(0) = 1. Besides, we make the following assumption instead of (V 1 ): We next give the following notations. As usual, for 1 ≤ s < +∞, we let Here we consider the following function space which is a Hilbert space endowed with the inner product , there exist constants γ s > 0 such that (2.2) Furthermore, we have the following compactness lemma due to [1].
We obverse that formally problem (1.1) is the Euler-Lagrange equation associated with the natural energy functional I : E → R given by It is well known that I may be not well defined in general in E. To overcome this difficulty, we make a change of variables constructed by Shen and Wang in [20], as Let us first collect some properties of the change of variables G : R → R, which will be used frequently in the sequel of the paper.
Therefore, after the change of variables, we obtain the following functional

HONGXIA SHI AND HAIBO CHEN
Note that the critical points of J are the weak solutions of the following equation (see [20]): a solution of (1.1).
Proof. First we prove that J is well defined in E. Since g is a nondecreasing positive function, we get We obtain that for each v ∈ E, It follows from the condition (f 1 ) that Thus, J is well defined in E.
Next we verify that J ∈ C 1 (E, R). Note that for any v, ψ ∈ E fixed, using the mean value theorem gives Since |t| ≤ 1, we get It is easily seen that Then by the Lebesgue dominated convergence theorem, we conclude that Similarly, using the properties of G −1 (v), the assumption (f 1 ) and the Lebesgue dominated convergence theorem lead to These indicate that J ∈ C 1 (E, R) and that for any ψ ∈ E, The proof is complete.
Recall that a sequence {v n } ⊂ E is said to be a (C) c -sequence if J(v n ) → c and (1 + v n )J (v n ) → 0, J is said to satisfy the (C) c -condition if any (C) c -sequence has a convergent subsequence. Proof. Let {v n } ⊂ E be a (C)-sequence for J at level c ∈ R, that is, Therefore, there is a constant C > 0 such that We first prove that there exists C 1 > 0 such that If it is not true, we suppose that v n Setting w n = G −1 (v n )/ v n 0 , then Passing to a subsequence, we may assume that w n w in E, w n → w in L s (R N ) for 2 ≤ s < 2 * , and w n → w a.e. on R N .
If w = 0, then meas(Ω) > 0, where Ω := {x ∈ R N : w = 0}. For x ∈ Ω, we have |G −1 (v n ))| → ∞ as n → ∞. Hence Ω ⊂ Ω n (r 0 , ∞) for large n, where r 0 is given in (f 2 ). By (f 2 ), we obtain Hence, using Fatou's lemma, we have It follows from (2.6) and (2.13) that which is a contradiction. Thus, the inequality Next, in order to show {v n } is bounded in E, we only need to prove that there exists (2.14) In fact, we may assume that v n = 0 (otherwise the conclusion is trivial). If this conclusion is not true, passing to a subsequence, we have

HONGXIA SHI AND HAIBO CHEN
Similar to the idea of [27], we assert that for each ε > 0, there exists C 7 > 0 independent of n such that meas(Ω n ) < ε, where Ω n := {x ∈ R N : |v n (x)| ≥ C 7 }. Otherwise, there is an ε 0 > 0 and a subsequence {v n k } of {v n } such that for any positive integer k, which is a contradiction. Hence the assertion is true. Notice that as |v n (x)| ≤ C 7 , by (6) and (7) in Lemma 2.2, we have Thus, (2.15) On the other hand, by virtue of the integral absolutely continuity, there exists ε > 0 such that whenever Ω ∈ R N and meas(Ω ) < ε, Combining (2.15) with (2.16), we have which implies that 1 ≤ 1 2 , a contradiction. This implies that (2.14) holds. Hence {v n } is bounded in E. The proof of this lemma is now finished. Proof. Indeed, by the boundedness of {v n } and the compactness of the embedding E → L s (R N ) (2 ≤ s < 2 * ), up to a subsequence, we have v n v in E, v n → v in L s (R N ) for all 2 ≤ s < 2 * and v n → v a.e. on R N . First, we claim that there exists C 10 > 0 such that (2.17) Indeed, we may assume v n = v (otherwise the conclusion is trivial). Set .

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Argue by contradiction and assume that is strictly increasing, and for each C 11 > 0 there exists δ 1 > 0 such that d ds Hence, we see that h n (x) is positive. Hence, By a similar argument as (2.15) and (2.16), we can conclude a contradiction.
On the other hand, we have By virtue of (2.17) and (2.18), we have which implies that v n → v in E. This completes the proof.
To complete the proof of our theorem, we state the following symmetric Mountain Pass Theorem. 3. Proof of main results. Let {e j } be a total orthonormal basis of E and define Then E = Y m Z m and Y m is finite dimensional. Similar to Lemma 3.8 in [26], we have the following lemma.
Before going further, we need to show that there exists C 13 > 0 such that where S ρ := {v ∈ E : v = ρ}. Indeed, by a similar argument as (2.14), we can get this conclusion. Furthermore, by Lemma 3.1, we can choose an integer k ≥ 1 such that v 2 2 ≤ Lemma 3.2. Suppose that (g), ( V 1 ), (V 2 ) and (f 1 ) are satisfied, then there exist positive constants ρ and α such that Proof. For any v ∈ Z k with v = ρ < 1, since p ∈ (2, 2 * ), by (3.1), (3.2) and Lemma 2.2 (2), we have This completes the proof. Proof. For any finite dimensional subspace E ⊂ E, there is a positive integral number m such that E ⊂ Y m . Suppose to the contrary that there exist {v n } such that {v n } ⊂ E ⊂ Y m and v n → ∞, but 1 2 R N |∇v n | 2 dx + 1 Jointly with Lemma 2.2 (2), we have On the other hand, set w n = v n / v n , then, up to a subsequence, we can assume that w n w in E, w n → w in L s (R N ) for all 2 ≤ s < 2 * and w n → w a.e. on R N . Let A = {x ∈ R N : w(x) = 0} and B = {x ∈ R N : w(x) = 0}.
By (f 1 ) and (f 2 ), there exists C 14 > 0 such that Since w n → w in L 2 (R N ), it is clear that which contradicts with (3.4). This shows that meas(A) = 0, i.e., w(x) = 0 a.e. on R N . According to the fact that all norms are equivalent on the finite dimensional space and the Sobolev embedding theorem, we have 0 = lim n→∞ w n s ≥ C lim n→∞ w n = C > 0.
This is a contradiction. This completes the proof.
Proof of Theorem 1.1. Set Φ = J, X = E, Y = Y m , Z = Z m in Theorem 2.6. Obviously, J(0) = 0 and (f 4 ) implies that J is even. By Lemmas 2.5, 3.2 and 3.3, all conditions of Theorem 2.6 are satisfied. Thus, problem (2.5) possesses infinitely many nontrivial solutions sequence {v n } such that J(v n ) → ∞ as n → ∞. Namely, problem (1.1) also possesses infinitely many nontrivial solutions sequence {u n } such that I(u n ) → ∞ as n → ∞.