Existence and concentration of positive solutions for a class of discontinuous quasilinear Schrödinger problems in RN$\mathbb{R}^{N}$

where > 0, W is a given potential, k ∈ R, g and h are real functions. Equation (1.1) with various types of h appears in several areas of physics. For example, in the case h(s) = (1 + s) 2 , problem (1.1) models the self-channeling of a high-power ultra-short laser in matter, the propagation of a high-irradiance laser in a plasma creates an optical index depending nonlinearly on the light intensity and this leads to interesting new nonlinear wave equations [6, 7]. For more applications, we can refer to [8–10] and the references therein. Here, we are interested in studying the case h(s) = s, which is used to model a superfluid film in plasma physics [4], especially the existence of standing wave solutions, that is, solutions of type ψ = exp(–iEt/ )u(x) with E ∈R and function u > 0 [11–13]. After a direct computation, problem (1.1) is equivalent to


Introduction
Recently many papers [1][2][3][4][5] have focused on studying the existence of solutions for the following quasilinear Schrödinger equations: where > 0, W is a given potential, k ∈ R, g and h are real functions. Equation (1.1) with various types of h appears in several areas of physics. For example, in the case h(s) = (1 + s) 1 2 , problem (1.1) models the self-channeling of a high-power ultra-short laser in matter, the propagation of a high-irradiance laser in a plasma creates an optical index depending nonlinearly on the light intensity and this leads to interesting new nonlinear wave equations [6,7]. For more applications, we can refer to [8][9][10] and the references therein. Here, we are interested in studying the case h(s) = s, which is used to model a superfluid film in plasma physics [4], especially the existence of standing wave solutions, that is, solutions of type ψ = exp(-iEt/ )u(x) with E ∈ R and function u > 0 [11][12][13]. After a direct computation, problem (1.1) is equivalent to It is well known that there exist lots of results on discussing Eq. (1.2) with k = 0, i.e., the following semilinear case: In [14] Rabinowitz used the mountain pass theorem to prove the existence of positive solutions of (1.3) for > 0 and V satisfying (V0) V ∞ = lim inf |x|→∞ V (x) > inf x∈R N V (x) = m > 0. Later, Alves and Figueiredo [15] extended (1.3) to the p-Laplace case with 2 ≤ p < N and proved that these solutions concentrate at global minimum points of V ( x) as → 0. More results can be found in [16][17][18][19][20] and so on.
Compared to the semilinear case, the quasilinear case (k = 0) becomes much more complicated as there is no suitable space for the energy functional corresponding to problem (1.2) for N ≥ 2. In order to overcome this difficulty, in [21], by changing of variables, the authors reduced the quasilinear equation (1.2) into the semilinear case. Based on this fact, problem (1.2) has been widely studied by assuming different hypotheses on V and f . Moameni [22] obtained the existence of a positive solution by assuming that f is a nonnegative function for N ≥ 2, and the potential function V is radially symmetric. Miyagaki and Moreira [11] derived the existence and multiplicity of solutions for problem (1.2) when the nonlinearity is indefinite in sign. Liu et al. [12,13] developed a perturbation method, the main idea of which is adding a regularizing term to recover the smoothness of the energy functional, so that the standard minimax theory can be used. Utilizing this method and a constrained minimization argument, they proved that problem (1.2) has a positive solution. Later, Wu [23] showed the existence of high energy solutions by employing the perturbation method for a general quasilinear problem. Recently, Carrião et al. [1] investigated the existence of a least energy solution for a class of nonhomogeneous asymptotically linear Schrödinger equations in R N via the Pohozaev manifold. It is worth to point out that different from semilinear problems, the critical exponent of problem (1.2) is 22 * , not 2 * , where 22 * = 4N N-2 . This will lead to some difficulties. For example, some properties in the usual Sobolev space cannot be used directly. The behavior of h at infinity plays an important role when searching for a solution to problem (1.2), mainly supercritical, critical or subcritical cases, where h behaves at infinity as |s| r-1 s, with r + 1 > 22 * , r + 1 = 22 * or r + 1 < 22 * , respectively. The critical case of (1.2) was considered in [24][25][26][27]. The supercritical results can be found in [28][29][30][31][32][33][34][35][36][37][38][39] and the references therein.
However, there seems to be little progress on the existence of positive solutions for general quasilinear elliptic equations with discontinuous nonlinearity. Based on this fact, we will study the quasilinear Schrödinger Eq. (1.2) from a discontinuous point of view. To some degree, the discontinuous case is more suitable to objective reality, and a smooth situation is usually just an ideal case. Hence, we consider the existence and concentration of solutions for the following problem: (1.4) where , β > 0 are positive parameters, p ∈ (3, 22 * -2) if N ≥ 3 or p ∈ (3, +∞) if N = 1, 2, V ∈ C(R N , R + ) satisfying (V0).
As is well known, the interest in studying nonlinear partial differential equations with discontinuous nonlinearities has increased since many free boundary problems and obstacle problems may be reduced to partial differential equations with nonsmooth potentials. Among these problems, we have the seepage surface problem, the obstacle problem, and the Elenbaas equation, see [40][41][42]. The area of nonsmooth analysis is closely related with the development of critical point theory for nondifferentiable functionals, in particular, for locally Lipschitz continuous functionals based on Clarke's generalized gradient [43]. In 1981, Chang [40] extended the variational method to a class of nondifferentiable functionals, and directly applied the variational method to prove some existence of theorems for PDE with discontinuous nonlinearities. It provides an appropriate mathematical framework to extend the classic critical point theory for C 1 -functionals in a natural way, and to meet specific needs in applications, such as nonsmooth mechanics and engineering. For a comprehensive understanding, we refer to Refs. [44][45][46][47][48][49][50][51][52][53].
This paper mainly discusses the existence of positive solutions to problem (1.4). Contrast to the previous results, our methods are totally different from those used in previous papers, since we are dealing with a discontinuous and non-convex problem. The main differences are the following: (1) Unlike [1], the lack of differentiability of nonlinearities causes some technical difficulties. This means that variational methods for C 1 functionals are not suitable in our case, since in our case, the energy functional is only locally Lipschitz continuous. Therefore, we have to use another variational approach based on the nonsmooth critical point theory due to Clarke [43] and Chang [54]. In contrast to C 1 variational methods, this method is not adequately developed, and we need to improve it. (2) In [1], if the energy functional associated to problem (1.2) is differentiable, it can be discussed on the Nehari manifold and the mountain pass level is equal to the minimum of the energy functional on Nehari manifolds, which is a key point in lots of papers. However, all these properties are not true for nondifferentiable problems. Hence, the arguments used in the above references cannot be directly repeated and we need to develop some new techniques to get over these difficulties. (3) Due to the appearance of the non-convex term (u 2 )u, some arguments used in standard semilinear problems cannot be used, therefore lots of estimates in this paper need to be reestablished.
is not compact, and the compact embedding is very crucial to deduce (PS) sequences in variational methods, we have to use other means to overcome this difficulty. The main result is the following. Our paper is organized as follows. In Sect. 2, we give some basic results involving locally Lipschitz continuous functionals. In Sect. 3, we deal with the existence of solutions for an auxiliary problem. Then we prove Theorem 1.1 in Sect. 4.

Preliminary results
In the sequel, we will use the following basic notations.
• means weak convergence while → means strong convergence. • C and C i (i = 1, 2, . . .) denote estimated constants (the exact value may be different from line to line). o n (1) denotes a sequence whose limit is 0 as n → ∞. • (X, · ) denotes a (real) Banach space and (X * , · * ) denotes its topological dual, | · | r denotes the norm of L r (R N ).
It is easy to see that the function ν → I 0 (u; ν) is sublinear, continuous and so is the support function of a nonempty, convex and w * -compact set ∂I(u) ⊂ X * , defined by If I ∈ C 1 (X), then Clearly, these definitions extend those of the Gâteaux directional derivative and gradient.

Definition 2.3 ([46])
(i) I satisfies the nonsmooth (PS) c condition if every sequence {u n } ⊂ X satisfying has a strongly convergent subsequence, where m I (u n ) = inf u * n ∈∂I(u n ) u * n X * . (ii) I satisfies the nonsmooth C-condition if every sequence {u n } ⊂ X satisfying I(u n ) → c and 1 + u n m I (u n ) → 0, has a strongly convergent subsequence, where m I (u n ) = inf u * n ∈∂I(u n ) u * n X * .
, z X for all u, z ∈ X; (iv) (Lebourg's mean value theorem) Let u and v be two points in X. Then there exists a point ξ in the open segment between u and v, and u * ξ ∈ ∂h(ω) such that (v) (Second chain rule) Let Y be a Banach space and j : Y → X be a continuously differentiable function. Then h • j is locally Lipschitz and a.e. in R N .
Assume that for each f ∈ Γ , there is some Then there exists a sequence u n ∈ X satisfying

An auxiliary problem
In this section, we firstly discuss an auxiliary problem, which is very important in proving Theorem 1.1. Note that weak solutions of (1.4) are critical points of the following functional: While, in order to find critical points of (3.1), we need to study the existence of solutions to problem (1.4) with = 1, i.e., The Euler-Lagrange functional corresponding to problem (3.2) I a : E → R is given by In order to overcome this difficulty, we adopt an method developed by Liu et al. [56] and Colin and Jeanjean [21]. Make the change of variables by From [21], one has the following lemma.

Lemma 3.1
The function f (t) and its derivative satisfy the following properties: Therefore, after the change of variable, from I a (u) we have the following functional where J a is well defined on the space E. Arguing as in [21], if v is a critical point of the functional J a , then u = f (v) is a critical point of the functional I a , i.e., u = f (v) is a solution of problem (3.2). Since we are looking for positive solutions to problem (3.2), we only need to require f (v) > 0, i.e., v > 0.

Lemma 3.2
The functional J a satisfies the mountain pass geometry.
Proof We introduce the following notations for the functional J a : Noting that p > 3, there exist r, β > 0 such that where ϕ 0 = max{a, 1}. Hence for t 0 > 0 sufficiently large, we obtain e = t 0 ϕ satisfying J a (e) < 0 with e ∈ E \ S r (0).
Note that J a (0) = 0, then J a satisfies the mountain pass geometry. It follows from the above lemma and Lemma 2.1 that there exists a sequence {v n } ⊂ E satisfying where c a is the mountain pass level of the functional J a .
Next, we will prove that {v n } given in (3.7) is bounded in E.

Lemma 3.3 The sequence {v n } is bounded in E.
Proof By (3.7) we have By Proposition 2.2, one has Hence From (3.8) and (3.9) we have which means that v n 0 is bounded. Using the same arguments used in [57, Lemma 2.1] we can obtain that v n is bounded in E, which completes the proof.
The following lemma is a key point in our analysis because the functional Q 2 is not compact. For each R > 0, let Q 2,R : L p+1 (B R (0)) → R be the function Furthermore, for each ϕ ∈ L p+1 (B R (0)), define the functionφ ∈ L p+1 (R N ) bỹ

Lemma 3.4 Let {v n } ⊂ E with v n v in E and
Proof Firstly, we denote by v n,R , γ n,R , v R and γ 0,R the restriction of the functions v n , γ n , v and γ 0 to B R (0). For ∀ϕ ∈ L p+1 (B R (0)), from a simple computation one has Noting that which means γ n,R ∈ ∂Q 2,R (v n,R ).
Employing the fact that R > 0 is arbitrary, we have a.e. in R N . and v n → v in L q loc R N for q ∈ 1, 2 * . (3.11)

Claim 1 The weak limit v is nontrivial.
In fact, if v ≡ 0, the limit v n → 0 in E does not hold as c a > 0. From Lions lemma [58], there exist {y n } ⊂ R N and α, r > 0 satisfying Since we are assuming v = 0, from the Sobolev embedding theorem we obtain that {y n } is unbounded. Now set w n (x) = v n (x + y n ). (3.12) Employing the boundedness of {v n } in E, we infer that {w n } is bounded in E. Thus, there exist w ∈ E \ {0} and subsequence of {w n }, still denote by itself, such that w n w in E (3.13) and where γ n ∈ ∂Q 2 (v n ), and so By Fatou's lemma, we have Passing to the limit of R → +∞, from the above inequality one deduces that (3.14) Once we have w = 0, there exists t > 0 such that tw ∈ N ∞ , where N ∞ is the Nehari manifold associated with J ∞ defined by i.e., from which it follows that c ∞ ≤ J ∞ (tv). Consequently According to Fatou's lemma and the inequality g(f (s))f (s)s ≥ (p + 1)G(f (s)) for all s ≥ 0, we derive that that is, which is a contraction. Hence v ≥ 0 and v = 0.
In the following, we will prove that v is a solution of problem (3.2). With this aim in mind, we need to show Furthermore, by Lemma 3.4 we have which means that v is a nonnegative weak solution of the following problem: Hence (3.19) and (3.20) mean that v is a weak solution of problem (3.2).
Remark 3.1 Due to the fact that V (x) ≥ m for all x ∈ R N , it is easily to verify, by using the Stampacchia theorem, that {x ∈ R N : f (v(x)) = a} has null measure for a small enough. Thus the weak solution v satisfies This is very important in many applications.

Existence and concentration of solution for (1.4)
In this part, we define the space Similar to (3.2), the dual energy functional associated with (1.4) is defined by and c ,a denotes its mountain pass level. Now, we are ready to prove Theorem 1.1.
Proof of Theorem 1.1 Divide the proof into two steps.
For each R > 1, we denote by ϕ R andṽ R the functions A direct computation shows that v R →ṽ in H 1 R N as R → +∞.
Thusṽ R = 0 for R sufficiently large. By this, there exists t R > 0 such that These facts mean that Once that c 0 < c ∞ (see [14]), we can choose δ, R > 0 such that and t > 0 satisfying J ,a (t * v k ) < 0 uniformly for , a > 0 small enough. Next, we considerγ (t) = t(t * v k ) for t ∈ [0, 1], whereγ ∈ Γ . By the definition of c ,a one has c ,a ≤ max for somet =t( , a, R) > 0.
For each given R > 0, it is obvious that there exist positive constants A 1 and A 2 such that A 1 ≤t ≤ A 2 for , a > 0 small enough. Note that m ≤ V (x) for all x ∈ R N . Then Without loss of generality, we suppose that V (0) = m. Hence, for each ζ > 0, there exists 0 > 0 such that 0 < V ( x)m < ζ for ∈ (0, 0 ) and x ∈ sup tv k = B 2R (0) from which one deduces that By the above inequality we have where C 1 , C 2 do not depend on , a > 0. Hence for , a > 0 small enough we have It follows from Theorem 3.1 that problem (1.4) has at least one nontrivial solution for , a > 0 sufficiently small.
Step 2. Now, we begin to prove the concentration of the solution. Denote by v ,a the solution given by step 1. Thus, there is γ ,a ∈ L p+1 p (R N ) such that v ,a (x) = 1 ,ḡ(f (v ,a (x)))f (v ,a (x))] a.e. in R N . Now, fix n → 0, a n → 0. v n = v n ,a n and γ n = γ n ,a n . We are ready to discuss the behavior of the maximum points related to {v n }, more precisely, if y n ∈ R N denotes a maximum point of v n , we will show that lim n→∞ V ( n y n ) = m.
By just the same method as used in (4.2) and (4.3), we obtain lim n→∞ c n ,a n = c 0 > 0. (4.5) Claim 2 There exist {z n } ⊂ R N and η, r > 0 such that In fact, if the claim does not hold, from a result due to Lions, one has lim n→∞ R N |v n | q dx = 0 for q ∈ (2, 2 * ). This limit combined with the fact that v n is a solution of (1.4) with = n and a = a n means that lim n→∞ c n ,a n = lim n→∞ J n ,a n (v n ) = 0, which contradicts (4.5).

Claim 3
The sequence w n = v n (·z n ) is strongly convergent in H 1 (R N ). Furthermore, lim |x|→∞ w n (x) = 0 uniformly in n ∈ N, that is, for ∀η > 0, there exists R > 0 such that Using the same arguments in Claim 1, we can assume that { n z n } is a convergent sequence in R N with n z n → z * ∈ V -1 (m). Moreover, we obtain that if w is the weak limit of {w n }, then In the following, we prove that lim |x|→∞ w n (x) = 0. (4.6) The main idea is borrowed from [15].
For each n ∈ N and L > 0, set where β > 1 is to be determined later. Take y L,n as a test function in (4.4), then For ξ sufficiently small, (4.7) and γ n ( For each > 0, by Young's inequality we have Taking ξ > 0 sufficiently small, the above inequality becomes We assert that v n ∈ L 2 * 2 2 (|x| ≥ R) for R large enough and uniformly in n. In fact, set β = 2 * 2 . By virtue of (4.10) one has |w L,n | 2 2 * ≤ Cβ 2 Observing that v n → v in H 1 (R N ), for R sufficiently large, we infer that |x|≥R/2 v 2 * n dx ≤ ε uniformly in n.
Choose η = δ * 2 and R > 0 such that w n (x) < δ * 2 ∀x ∈ R N \ B R (0) and n ∈ N, and so, if y n denotes a maximum point of w n , we derive w n (y n ) ≥ δ * and y n ∈ B R (0). Settingŷ n to be the maximum point of v n , we haveŷ n = y n + z n , which means ŷ n = n y n + n z n → z * . From the continuity of the function V one derives lim n→∞ V ( nŷn ) = V z * = m.
Thus the proof is completed.