Localized solutions of higher topological type for semiclassical generalized quasilinear Schrödinger equations

We consider the following semiclassical generalized quasilinear Schrödinger equation -ε2div(g2(v)∇v)+ε2g(v)g′(v)|∇v|2+V(x)v=f(v),x∈RN,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} -\varepsilon ^2\text {div} (g^2(v)\nabla v)+\varepsilon ^2g(v) g'(v)|\nabla v|^2+V(x)v=f(v), \quad x\in {\mathbb {R}}^{N}, \end{aligned}$$\end{document}where V∈C1(RN)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V\in C^1({\mathbb {R}}^N)$$\end{document} is bounded and f is odd in v and satisfies a monotonicity condition. We establish the existence of multiple localized solutions concentrating at the set of critical points of V.


Introduction
In this paper, we are concerned with the generalized quasilinear Schrödinger equations − ε 2 div(g 2 (v)∇v) + ε 2 g(v)g (v)|∇v| 2 where N ≥ 1. It is well known that the equation is related to the existence of standing wave solutions for several nonlinear or quasilinear equations in mathematical physics. The first example, the nonlinear Schrödinger equations iεz t = −ε 2 Δz + W (x)z − h(|z| 2 )z, where z is a complex wave function, W is a real potential function and h is a real function. Let z(t, x) = exp(−iEt/ε)v(x), with V (x) = W (x) − E, we get the nonlinear Schrödinger equations like this which is a special case of (1.1) as g(v) = 1. Equation (1.2) have been investigated widely by many researchers. To the best of our knowledge, the first result in this line is due to Floer-Weinstein [11]. These authors considered the one-dimensional case and constructed via a Lyapunov-Schmidt reduction which relies on the uniqueness and non-degeneracy of the ground state solution of the limiting equation, a family of solutions concentrating around any nondegenerate critical point of the potential V . Under the well-known Ambrosetti-Rabinowitz condition Wang [21] established that the solutions concentrate at global minimum points of V as ε → 0 with power-law nonlinearity; moreover, a point at which a sequence of solutions concentrates must be critical for V . The existence of positive solutions which concentrate around local minimum of V by introducing a penalization method was shown in [4]. As far as we know, there are few papers about multiple localized solutions concentrating around the set of critical points of V . In [8], as ε → 0, more and more sign-changing solutions clustered at a local minimum point of V were given by the Lyapunov-Schmidt reduction method. The existence of arbitrarily many nodal solutions and nodal solutions having arbitrarily many nodal domains was established in [23] on a bounded domain by the Lyapunov-Schmidt reduction method with adequate variational techniques. Under certain hypotheses on V and a general spectral assumption, the existence and multiplicity of semiclassical solutions were obtained for asymptotically linear nonlinearity via variational methods in [24]. The multiple localized solutions concentrating at the set of critical points of V were shown in [9]. The qualitative analysis of localized entire nodal solutions with power-type subcritical nonlinearity was given in [2]. They constructed an infinite sequence of localized sign-changing solutions which are obtained from a minimax characterization of higher-dimensional symmetric linking structure via the symmetric mountain pass theorem without using any non-degeneracy conditions.
The other examples include the equations They appear naturally in mathematical physics, for example in plasma physics and fluid mechanics, in the theory of Heisenberg ferromagnetism and magnons, in dissipative quantum mechanics and in condensed matter theory. For more details and references, see [14]. If we try to find the standing wave solutions which is a special case of (1.1) as , where l is a real function. As g 2 (v) := 1 + v 2 2(1+v 2 ) , i.e., l(s) = (1 + s) 1/2 , (1.5) changes to the equation which models the self-channeling of a high-power ultrashort laser in matter.
For (1.1) with ε = 1, there are much works studying the existence of solutions. The authors in [18] proved, through the change of variable, the existence of a nontrivial solution by the mountain pass theorem. Subsequently, in [5] the authors constructed nodal radial solutions with a radially symmetric potential. The existence of a positive solution with critical growth was shown in [6,7]. For the concentration behavior of solutions in (1.1), Li-Wu [13] studied the existence, multiplicity and concentration of solutions with critical growth by the Ljusternik-Schnirelmann theory. Recently, the existence and concentration behavior of ground state solutions was shown by [3]. To sum up, the study on the existence and multiplicity of localized solutions of (1.1) is for some given special functions f and g; a natural question is whether there is a unified approach to study (1.1) with general functions f and g? However, there seems to be little progress on the multiplicity of localized solutions for a generalized quasilinear Schrödinger equations (1.1) with general functions f and g. The present paper is devoted to this direction. The novelty of our main result is to weaken the (AR)-type conditions, establish multiple localized solutions of higher topological type concentrating around the set of critical points of V and extend to the generalized quasilinear Schrödinger equations.
is a consequence of (V 2 ) and obviously A is a compact subset of M. We assume that, without loss of generality, 0 ∈ A. For any set B ⊂ R N and δ > 0, define For any ε > 0 and any set A ⊂ R N , we also define The main result of this paper, which we prove in Sect. 3, is the following Remark 1.1. Assume g(t) = 1 and f (t) = t|t| 1+|t| , then an easy calculation implies that f satisfies (f 1 )-(f 5 ) with b < 1. The proof is simpler if the limit b 1 in (f 3 ) is infinity, so we omit it. Without loss of generality, assume that b 1 is finite. Remark 1.2. If g(t) = 1, then G(t) = t, we know from the assumptions (f 2 ) − (f 3 ) that the nonlinearity f is asymptotically linear, the generalized quasilinear Schrödinger equations (1.1) turn into the classical Schrödinger equations, see [2,9] for example. Remark that, in this case, the authors in [2] assumed that the nonlinearity satisfies the (AR) type condition (1.3), and the author in [9] considered asymptotically linear nonlinearity and obtained the multiplicity of localized solutions concentrating at the set of critical points of V . In our paper, the generalized quasilinear Schrödinger equations (1.1) we considered are more complicated. The aim of our paper is to weaken the (AR)-type conditions and extend some results in [2,9] to the generalized quasilinear Schrödinger equations. This paper is organized as follows. In Sect. 2, some preliminaries are given and we define an auxiliary problem by using the variant of the penalization argument introduced in [4]. Section 3 is devoted to Theorem 1.1; more precisely, we showed the existence of solutions for the auxiliary problem and then proved the solutions of the auxiliary problem are ones of the original problem. The concentration behavior was also considered. Notation. C, C 1 , C 2 , . . . will denote various positive constants whose exact value is inessential.
and S is the unit sphere in E.

Preliminary results
The hypotheses (V 1 ) and (V 2 ) imply that there is δ 0 > 0 such that 0 ∈ A ⊂ M 2δ0 and Then and According to (f 1 ) and (g), we have that f (t) ≥ 0 for t > 0 and f (t) ≤ 0 for t < 0. These lead to F (t) ≥ 0 andF (t) ≥ 0 for all t ∈ R. (2) and (3) follow from (f 2 ), (f 4 ) and the expression off . Now we try to find solutions of the following equation .
where V ε (x) := V (εx). The energy functional corresponding to (2.4) is formally the following Euler-Lagrange equation Consider a change of variable introduced by [18], u = G(w) = w 0 g(t) dt; then we have the following functional which is well defined on E and belongs to C 1 under the condition (g), Lemmas 2.2 and 3.1 below. It is easy to see that the critical points of I ε are weak solutions of the problem and if u ∈ E satisfies the above equation, then w = G −1 (u) ∈ E is a solution of (2.4). Let us collect some properties of g and G which have been proved in [18] and Lemma 2.1 in [6] except (5).
Define the Nehari manifold by Under our assumptions, we cannot make sure whether or not N ε is of class C 1 , so we take the methods developed by [10].

Proof of Theorem 1.1
Proof. (1) By Lemma 2.1-(1) and the expression of H, we finish the conclusion. (2) It follows immediately from (f 1 ), (f 2 ) and (f 3 ). Let It follows from the definition of ν and (V 1 ) that u = 0, a contradiction.
It follows from (f 2 ) and Lemma 2.
So, h u (t) has a positive maximum at t u > 0. Note also that h u (t) = I ε (tu), u = 0 is equivalent to We have by (g) and Lemma 2.
The assumption (f 4 ) implies that h(εx,G −1 (s)) g(G −1 (s))s is increasing for s > 0. Hence, t u is the unique positive number such that h u (t u ) = 0.
Proof. Without loss of generality, we may assume that V ⊂ S. Suppose by contradiction that there exists u n ∈ V and w n = t n u n such that u n → u ∈ F ε , I ε (w n ) ≥ 0 and t n → ∞ as n → ∞, where t n is as in Lemma 3.2-(1). Since |w n (x)| → ∞ for u(x) = 0, using Lemma 2.2-(4), (f 3 ) and the Lebesgue dominated convergence theorem we deduce Proof. Assume (u n ) ⊂ N ε is a Palais-Smale sequence for I ε , that is, I ε (u n ) → 0 and I ε (u n ) ≤ d for some d > 0.

3.3) a direct calculation yields
The second inequality above follows from (f 3 ) and small ε > 0. So, (u n ) is bounded in E and u n u in E up to a subsequence.
Next we will prove u n → u in E. Indeed, it suffices to show that for each δ > 0 there is R > 0 such that According to Lemma 2.2-(5), (V 1 ) and Lemma 2.1-(2), We finish the proof by taking sufficiently large R and n.
Let V ε := F ε ∩ S (recall that S is the unit sphere in E) and define the mapping m : Lemma 3.6. Assume w n ∈ V ε , w n → w 0 ∈ ∂V ε and m(w n ) = t n w n . Then I ε (t n w n ) → ∞ as n → ∞.

Proof. Note that
So, I ε (tw 0 ) → ∞, as t → ∞ by (f 5 ) and the Lebesgue dominated convergence theorem. For every C > 0, take t > 0 such that I ε (tw 0 ) ≥ C. Therefore, by Lemma 3.2-(1) Lemmas 3.7 and 3.8 below are taken from [20] (see Proposition 8 and Corollary 10 there). That the hypotheses in [20] are satisfied is a consequence of Lemmas 3.2-3.4 above with V ε as a proper open subset of S (see also [10], Sect. 4). In fact, assume h(t) := I ε (tw) and w ∈ S, h (t) > 0 for 0 < t < t w and h (t) < 0 for t > t w by Lemma 3.2, t w ≥ δ > 0 by Lemma 3.3 and t w ≤ R for w ∈ V ⊂ S by Lemma 3.4.

Lemma 3.7.
The mapping m is a homeomorphism between V ε and N ε , and the inverse of m is given by m −1 (u) = u u . We shall consider the functional Ψ ε : V ε → R given by Ψ ε (w) := I ε (m(w)).
There is a small ball B r (0) ⊂ M δ0 since 0 ∈ A ⊂ M 2δ0 . Let and Let S k be the unit sphere in H 1 0 (B r/ε k (0)). Taking a similar argument as Lemmas 3.2-3.8 (see also [10,20]), let U k := E k ∩ S k and define the mappingm : U k → N k by settingm(w) :=t w w, where N k is the Nehari manifold corresponding to I k andt w is the global maximum point of I k (tw) for t > 0. Let Ψ k : U k → R be given by Ψ k (w) := I k (m(w)), where Γ j = {A ∈ Σ k : γ(A) ≥ j}. It follows from the Index theory for even functional that I k has at least k critical values c j such that 0 < c 1 ≤ c 2 ≤ · · · ≤ c k .
We finish the proof.
then u 1 j,ε → 0 in L p (R N ) for 2 < p < 2 * by P.L. Lions' lemma. We observe that I ε (u j,ε ), u 1 j,ε = 0 and (3.5) imply The third integral on the left-hand side above tends to zero by (g), Lemma 2.2-(2) and the weak convergence. A similar argument as Proposition 7 in [1] implies |u j,ε | L ∞ (R N ) ≤ C for small ε > 0 by Lemma 3.10. According to Proposition 4.1 and (3.6), we obtain It follows from (g) and Lemma 3.1-(2) that the integral above on the right-hand side goes to zero. Hence, u 1 j,ε → 0 in E by Lemma 2.2- (5) and (V 1 ), and we are done. If j,ε → y 1 j up to a subsequence. Note that y 1 j may be infinity at this moment. Let w 1 j,ε = u 1 j,ε (· + y 1 j,ε ). Then w 1 j,ε u 1 = 0 in E up to a subsequence. Similarly as (3.6) and (3.7), since and moreover similarly as in the proof of Lemma 3.10, there is ρ 1 > 0 such that u 1 ≥ ρ 1 . If y 1 j ∈ M δ0/4 , it follows from (3.8) and the definition of χ that ) .
Using Lemma 2.1-(2), Lemma 2.2-(5) and (V 1 ), a contradiction with u 1 ≥ ρ 1 > 0. Hence, y 1 j ∈ M δ0/4 ⊂ M and y 1 j,ε ∈ M ε for small ε > 0. Employing a similar argument above and iterating it, the procedure has to stop in finite steps, since u j,ε ≤ η j by Lemma 3.10 and u j ≥ ρ 1 . Indeed, similarly as [22] u i j,ε Assume for each i, up to a subsequence, lim n→∞ ε n y i j,εn exists with lim n→∞ ε n = 0. Denote the set of these limiting points by for some 1 ≤ s j ≤ m j . Let for s j = 1.
Lemma 3.12. For 0 < δ < ϑ 1 , there exist C > 0 and c > 0 independent of n such that, for every 0 ≤ i ≤ m j and large n, Proof. We employ a similar argument as Lemma 4.5 in [2]. Let A similar argument as Proposition 7 in [1] implies |u j,εn | L ∞ (R N ) ≤ C for small ε n > 0 by Lemma 3.10.