Infinitely many solutions to quasilinear Schrödinger equations with critical exponent

This paper is concerned with the following quasilinear Schrödinger equations with critical exponent: −∆pu + V(x)|u|p−2u− ∆p(|u|)|u|u = ak(x)|u|q−2u + b|u|2ωp −2u, x ∈ RN . Here ∆pu = div(|∇u|p−2∇u) is the p-Laplacian operator with 1 < p < N, p∗ = Np N−p is the critical Sobolev exponent. 1 ≤ 2ω < q < 2ωp, a and b are suitable positive parameters, V ∈ C(RN , [0, ∞)), k ∈ C(RN , R). With the help of the concentration-compactness principle and R. Kajikiya’s new version of symmetric Mountain Pass Lemma, we obtain infinitely many solutions which tend to zero under mild assumptions on V and k.


Introduction and main result
In this paper, we establish the existence of infinitely many solutions which tend to zero for the following quasilinear Schrödinger equations with critical exponent The energy functional associated with (1.1) is given by (1.2) Here ∆ p u = div(|∇u| p−2 ∇u) is the p-Laplacian operator with 1 < p < N, p * = N p N−p is the critical Sobolev exponent.1 ≤ 2ω < q < 2ωp, a and b are positive parameters.V(x) and k(x) are continuous and satisfy the following conditions: (V) V ∈ C(R N , [0, ∞)) satisfies inf x∈R N V(x) ≥ V 0 > 0, and for each M > 0, meas{x ∈ R N : V(x) ≤ M} < +∞, where V 0 is a constant and meas denotes the Lebesgue measure in R N .
In recent years, a great attention has been focused on the study of solutions to quasilinear Schrödinger equations.Such equations arise in various branches of mathematical physics.For example, when p = 2, ω = 1, the solutions of (1.1) are related to the existence of solitary wave solutions for quasilinear Schrödinger equations where Ψ : R × R N → C, W : R N → R is a given potential, κ, h are real constants and ρ, h are real functions.This type of equations appear more naturally in mathematical physics and have been derived as models of several physical phenomena corresponding to various types of ρ(s).In the case ρ(s) = s, (1.3) was used for the superfluid film equation in plasma physics by Kurihara in [12] and [13].In the case ρ(s) = (1 + s) 1/2 , (1.3) models the self-channeling of a high-power ultrashort laser in matter (see [4,6]).Considering the case ρ(s) = s α , κ > 0 and putting Ψ(t, x) = exp(− iFt h )u(x), F ∈ R is some real constant, it is clear that Ψ(t, x) solves (1.3) if and only if u(x) solves the following elliptic equation: where we have renamed W(x) − F to be V(x).
For the case ακ = 1, h(x, u) = θ|u| p−1 u, Poppenberg, Schmitt and Wang in [19] studied the equation (1.4) by translating it into an ODE and then a ground state solution u ∈ W 1,2 (R) of problem (1.5) was obtained.They also got that the equation (1.5) admits a positive solution u ∈ W 1,2 (R) for any arbitrarily large values of θ.Later, Liu, Wang and Wang in [17] established the existence of ground states of soliton-type solutions for (1.4) as in the case α = 1, κ = 1 2 by the variational methods.Using a constrained minimization argument, Liu, Wang and Wang in [16] established the existence of a positive ground state solution for (1.4).As we know, Nehari method is used to get the existence results of ground state solutions in [10] and the problem is transformed to a semilinear one in [2,9] by a change of variables.Recently, the author in [23] studied the equation (1.4) and obtained that it has a positive and a negative weak solution under proper conditions of α, V, g.A natural question is that weather there exist infinitely many solutions for equations like (1.4).The authors in [7,8] investigated the following type quasilinear elliptic equation: Problem (1.6) can be reduced to the following quasilinear elliptic equations: By using the Pohozaev identity, the author has the nonexistence result for (1.7).
To the best of our knowledge, the existence of nontrivial radial solutions for (1.4) with g(x, u) = µu 2(2 * )−1 was firstly studied by Moameni in [18], where the Orlicz space as the same as it was used in [17].However, it seems that there is almost no work on the existence of infinitely many solutions to the quasilinear Schrödinger problem in R N involving critical nonlinearities and generalized potential V(x).
Motivated by the above discussions, the main goal of this paper is to study the existence of infinitely many solutions which tend to zero to the problem (1.1).The lack of compactness of the embedding from W 1,p (R N ) into L p * (R N ) prevents us from using the variational methods in a standard way.To overcome the lack of compactness caused by the Sobolev embeddings in unbounded domains and the critical exponent, some new estimates for (1.1) are needed to be re-established.We apply Lions' concentration-compactness principle [14,15] to give a more detailed analysis for the compactness of our problem.Thanks to the new version of symmetric Mountain Pass Lemma in [11], we give the proof of our main result.As far as we know, there are few results on this question, so the research in this paper is meaningful.Now we first give the definition of weak solutions for problem (1.1).
In the sequel we will omit the term weak when referring to solutions that satisfy the conditions of Definition 1.1.Our main result of this paper is stated as follows.
Theorem 1.2.Suppose that (V) and (K) hold, 1 ≤ 2ω < q < 2ω p. Then (i) ∀b > 0, ∃ a 0 > 0 such that if 0 < a < a 0 , problem (1.1) has a sequence of solutions {u n } with I(u n ) < 0, I(u n ) → 0 and lim n→∞ u n = 0.The outline of this paper is as follows.Reformulation of the problem and some preliminaries are given in the forthcoming section.In Section 3, behavior of (PS) sequences are established.The proof of Theorem 1.2 is given in Section 4.
We denote that L p (R N ) is the usual Lebesgue space with the norm is the best Sobolev constant.Various positive constants are denoted by C and C i .

Reformulation of the problem and preliminaries
The purpose of this section is to establish the variational structure of (1.1) and the main difficulty arises from the function space where the energy functional (1.2) is not well defined in W 1,p (R N ).For example, if 1 < p < N and u is defined by To overcome this difficulty, we employ an argument developed by Liu, Wang and Wang in [17] or Colin and Jeanjean in [5].We use the change of variables v = f −1 (u), where f is defined by , and f (0 The following result is due to Adachi and Watanabe in [1] which collects some properties of f . Lemma 2.1.The function f (t) enjoys the following properties: (1) f is uniquely defined C ∞ function and invertible. ( ( → a > 0 as t → +∞. ( After the above change of variables, we can rewrite our energy functional (1.2) in the terms of v: We first give the proof of the following weakly continuous lemma.
Lemma 2.2. (i Proof.Firstly, by ( 3) and ( 4) in Lemma 2.1, it is clear that F (v) and G(v) are well defined on W 1,p (R N ).Next, we prove that F (v), G(v) ∈ C 1 (R N ).It suffices to show that both F (v) and G(v) have continuous Gateaux derivatives on W 1,p (R N ).We only prove that F (v) has continuous Gateaux derivatives on W 1,p (R N ) since the case of the proof for G(v) is simpler.
Our proof is the same as the proof of Lemma 3.10 in [22] , for the convenience of the readers, we present the process.Let v, g ∈ W 1,p (R N ).Given 0 < |t| < 1, by the mean value theorem, there exists λ ∈ (0, 1) such that where the conclusions of Lemma 2.1 (2) and ( 5) are used.By the Hölder inequality and assumption of (K), we have It follows from the Lebesgue Dominated Convergence Theorem that F (v) is Gateaux differentiable and Now, we give the proof of continuity of Gateaux derivative.Assume that . Using the Hölder and Sobolev inequalities, we have From the above analysis we can get that J(v) is well defined on W 1,p (R N ) under the assumptions of (V) and (K).The standard arguments applied in [20,22] show that J(v) belongs to C 1 (W 1,p (R N ), R).As in [5], we note that if v is a nontrivial critical point of J, v then is a nontrivial solution of the problem 1 p , we conclude that u is a nontrivial solution of the problem (1.1).Now we can restate Theorem 1.2 as follows.
Theorem 2.3.Suppose that (V) and (K) are held, ω > 1/2, 2ω < q < 2ω p. Then (i) ∀b > 0, ∃ a 0 > 0 such that if 0 < a < a 0 , problem (2.1) has a sequence of solutions {v n } with J(v n ) < 0, J(v n ) → 0 and lim n→∞ v n = 0. (ii 3 Properties of (PS) c sequences In this section, we perform a careful analysis of the behavior of minimizing sequences with the aid of Lions' concentration-compactness principle [14,15], which allows us to recover the compactness below some critical threshold.
Let E be a real Banach space and J : E → R be a function of class C 1 .We say that {v n } ⊂ E is a (PS) c sequence if J(v n ) → c and J (v n ) → 0. J is said to satisfy the Palais-Smale condition at level c ((PS) c for short) if any (PS) c sequence contains a convergent subsequence.Lemma 3.1.Assume (V) and (K), {v n } ⊂ W 1,p (R N ) be a (PS) c sequence for J at level c < 0 and 2ω < q < 2ω p. Then (i) there exists C > 0 such that, for all n ∈ N, and which implies that for n large enough, there exists C > 0 such that In the following, we need to show {v n } is bounded in W 1,p (R N ).From (3.6), we need to prove that R N V(x)|v n | p dx is bounded.By (V), and using Lemma 2.1 (6), ). Therefore we can assume that In view of the concentration-compactness principle [14,15], there exist a subsequence, still denoted by { f (v n )}, µ, ν ∈ M(R N ∪ {∞}) which are the positive finite Radon measures on R N ∪ {∞}, an at most countable set J , a set of different points {x j } ⊂ R N , and real numbers µ j , ν j such that the following convergence hold in the sense of measures From the above two equations and the Sobolev inequalities, it follows easily that Concentration at infinity of the sequence {u n } is described by the following quantities: We claim that J is finite and, for j ∈ J , either In fact, for ε > 0, letting x j be a singular point of the measures µ j and ν j , φ j (x) be a smooth cut-off function centered at x j such that 0 In the following we estimate each term in (3.8).By Lemma 2.1 (5) and the expression of f , we have Also we have and lim By the weak continuity of F (v), we get From (3.9)-(3.12),by the weak continuity of F , we have Combining with (3.7), we obtain either (i) which implies that J is finite.The claim is thereby proved.
To analyze the concentration at ∞, we follow closely the argument used in [21].By choosing a suitable cut-off function Similar to the process of (3.9), we can get Using the weak continuity of F , we have lim we get Next, we claim that (ii) and (iv) cannot occur if a and b are chosen properly.In fact, by (3.4) and (V), we have Then if (iv) holds, from the weak lower semicontinuity of the norm and the weak continuity of F , we have, This inequality implies that Therefore from (3.17) and (iv), (v) If A is compact, then γ(A) < +∞ and there exists δ > 0 such that N δ (A) ⊂ Σ and γ(N δ (A)) = γ(A), where N δ (A) = {x ∈ X : x − A ≤ δ}.
Thanks to the work of Kajikiya in [11], we take the following version of the symmetric mountain-pass lemma.Proposition 4.2.Let E be an infinite-dimensional space and J ∈ C 1 (E, R) and suppose the following conditions hold: (A 1 ) J(u) is even, bounded from below, J(0) = 0 and J(u) satisfies the local Palais-Smale condition (PS for short).
(A 2 ) For each k ∈ N, there exists an A k ∈ Σ k such that sup u∈A k J(u) < 0.
Then either (i) or (ii) below holds.
(i) There exists a sequence {u k } such that J (u k ) = 0, J(u k ) < 0 and {u k } converges to zero.
Remark 4.3.From Proposition 4.2 we have a sequence {u k } of critical points such that J(u k ) ≤ 0, u k = 0 and lim k→∞ u k = 0.
In order to get infinitely many solutions we need some lemmas.Let J(v) be the functional defined as before, 1 < 2ω < q < 2ωp, and a > 0, b > 0.Then, by (3.4), Since 1 < 2ω < q < 2ωp, it is easy to see that, for the given b > 0, we can choose small a * > 0 such that if 0 < a < a * , there exists 0 < t 0 < t 1 such that g(t) < 0 for 0 < t < t 0 ; g(t) > 0 for t 0 < t < t 1 ; g(t) < 0 for t > t 1 .
(3) ∀b > 0, ∃ a * = min{a * , a * } > 0 such that if 0 < a < a * , then J satisfies (PS) c .Proof.The aforementioned (1) and ( 2) are immediate.To prove (3) and ( 4), observe that all (PS) sequences for J with c < 0 must be bounded.Similar to the proof of Lemma 3.1, there exists a strong convergent subsequence in W 1,p (R N ).(3) or (4) above, it then follows from (PS) c that K c (c < 0) is compact.Lemma 4.6.Assume that (K) is held, then for the given n ∈ N, there exists n < 0 such that Proof.Let X n be a n-dimensional subspace of W 1,p (R N ).For any v ∈ X n , v = 0, write v = r n w with w ∈ X n , w = 1 and then r n = v .From the assumption (K), it is easy to see that, for every w ∈ X n with w = 1, there exists d n > 0 such that R N k(x)|w| q 2ω dx ≥ d n .Thus for 0 < r n < t 0 and Lemma 2.1 (2),(5), we have Therefore we can choose small r n ∈ (0, t 0 ) such that J(v) ≤ n < 0. Let S r n = {v ∈ X n : v = r n }.Then −∞ < c n ≤ n < 0 since J n ∈ Σ n and J is bounded from below.