GROUND STATE SOLUTIONS FOR ASYMPTOTICALLY PERIODIC QUASILINEAR SCHR¨ODINGER EQUATIONS WITH CRITICAL GROWTH

. In this paper, we are concerned with the existence of ground state solutions for the following quasilinear Schr¨odinger equation: where N ≥ 3, V, g are asymptotically periodic functions in x . By combining variational methods and the concentration-compactness principle, we obtain a ground state solution for equation (1) under a new reformative condition which unify the asymptotic processes of V,g at inﬁnity.


Introduction and main result. Quasilinear Schrödinger equations of the form
have been derived as models of several physical phenomena and have been the subject of extensive study in recent years, where W : R N → R is a given potential, κ is a positive constant and l, ρ are real functions. Here we consider the existence of standing wave solutions for quasilinear Schrödinger equations of form (2) with ρ(s) = s, κ = 1. Seeking solutions of the type stationary waves, namely, the solutions of the form ψ(t, x) = exp(−iEt)u(x), E ∈ R, we get an equation of elliptic type which has the formal structure: where V (x) = W (x)−E is the new potential function and K(x)|u| 22 * −2 u+g(x, u) = l(x, u 2 )u is the new nonlinearity.
For the critical case, we would like to mention [5,6,8,11,25,27,31,33] and the references therein. It seems that Moameni [27] first studied the critical case when the potential V is radial and satisfies some geometric conditions. In [8], they obtained a positive classical solution by using the concentration compactness principle of Lions [15]. In [5], by using a change of variables and minimization argument, they obtained a sign-changing minimizer with k nodes of a minimization problem. Reference [25] established the existence of both one-sign and nodal ground states of soliton type solutions by the Nehari method. The method is to analyze the behavior of solutions for subcritical problems to pass limit as the exponent approaches to the critical exponent. In [11], they got the existence, concentration and multiplicity of weak solutions by employ the minimax theorems and Ljusternik-Schnirelmann theory. For asymptotically periodic nonlinearities with critical exponent there are considerably fewer results, here we mention [33].
It is worth pointing out that the related semilinear equation with the asymptotically periodic condition has been extensively studied, see [13,17,18,34,37,38] and their references. In [13,34,37,38], they discussed the existence of solutions for problem (3) without the second order derivatives ∆(u 2 )u, when the problem is strongly indefinite, that is, 0 lies in a spectral gap of − + V . We would like to point out that in a recent paper [17] and [18], Liu et al. have given reformative conditions which unify the asymptotic processes of V, g at infinity. The asymptotic processes is weaker than those in [13,34,37,38]. We borrow an idea from [17] and [18] to obtain the ground state solution for problem (3). The only work with quasilinear asymptotically periodic Schrödinger equations with critical growth is our reference [33], they obtained only the existence of nontrivial solutions for problem (3) by using the mountain pass theorem. Here, we consider the ground state solution, which has great physical interests.
The main purpose of present paper is to establish the existence of a positive ground state solution for problem (3) and the corresponding periodic problem. There are several difficulties in our paper. The mainly one is the reformative condition which unifies the asymptotic processes of V, g at infinity. So we need rather careful estimates between g 0 and g, V 0 and V . Besides, the nonlinear term g in our paper need not be differentiable, then the constrained manifold need not be of class C 1 in our case. We should employ a similar argument in [17] to conquer it. At last, the possible lack of compactness due to the criticality of the growth and the unboundedness of the domain, in order to obtain the existence of the solutions we will turn to the concentration compactness lemma due to [15,16].
We suppose that V satisfies the following assumption: We also assume the following conditions on K : Because we look for a positive solution, we can assume that g(x, s) Now we state our main result: Suppose that (V ), (K) and (g 1 ) − (g 5 ) are satisfied, then problem (3) possesses a positive ground state solution.
As a by-product of our calculations we can obtain a weak solution for the periodic problem.
Remark 1. We compare our results with [33] as follows: (1) We find a ground state solution with the aid of a Nehari-type constraint, while in [33] the authors obtained a nontrivial solution by using the mountain pass theorem.
(2) We consider a new reformative condition which unify the asymptotic processes of V, g at infinity, which means F and F 0 contain more elements than those in [33].
(3)The aim of (g 3 ) is to obtain a ground state solution. In [33], the authors had employed other different types of conditions on g.

Notation:
In this paper, we use the following notations: • H 1 (R N ) is the usual Hilbert space endowed with the norm • L s (R N ) is the usual Banach space endowed with the norm u s s = R N |u| s dx, ∀s ∈ [1, +∞).
• u ∞ = ess sup x∈R N |u(x)| denotes the usual norm in L ∞ (R N ).
• |Ω| denote the Lebesgue measure of the set Ω.
2. Some preliminary results. We observe that formally problem (3) is the Euler-Lagrange equation associated with the energy functional From the variational point of view, the first difficulty we have to deal with problem (3) is to find an appropriate function space where the above functional is well defined. In the spirit of the argument developed in [3]. We make a change of After the change of variables from J, we obtain the following functional: Then I(v) = J(u) = J(f (v)) and I is well defined on E, I ∈ C 1 (E, R) under the hypotheses (V) and (g 1 ) − (g 5 ). Moreover, we observe that if v is a critical point of the functional I, then the function u = f (v) is a solution of problem (3) (see [3]). Below we summarize the properties of f , which have been proved in [3,33] and [7]. (1) f is uniquely defined, C ∞ and invertible; there exists a positive constant C such that |f (t)| ≥ C|t| for |t| ≤ 1 and |f (t)| ≥ C|t| 1/2 for |t| ≥ 1; (10) |f (t)f (t)| ≤ 1 √ 2 for all t ∈ R; (11) the function f (t)f (t)t −1 is strictly decreasing for t > 0; GROUND STATE SOLUTIONS FOR QUASILINEAR SCHRÖDINGER EQUATIONS 1125 (12) the function f p (t)f (t)t −1 is strictly increasing for p ≥ 3 and t > 0; (13) there exist constants M, R > 0 such that, for all t ≥ R, For any δ > 0, there exist r δ > 0, C δ > 0 and α ∈ (2, 2 * ) such that Proof. We only need to proof the first inequality (5). It follows from (g 1 ), (g 3 ) that G(x, s) ≥ 0 and for s > 0, Combining with Lemma 2.1-(6), one has The rest inequalities follow from (g 1 ) − (g 4 ) immediately. lim Lemma 2.4 ( [17]). Suppose that condition (V ) holds. Then there are two positive constants C 1 and C 2 such that

GROUND STATE SOLUTIONS FOR QUASILINEAR SCHRÖDINGER EQUATIONS 1127
Let By (g 3 ) and Lemma 2.1-(12), the function is strictly increasing for s > 0. Hence also s → Z(s) is strictly increasing according to Lemma 2.1-(11) (12). So there is a unique t u > 0 such that h (t u ) = 0. The conclusion is an immediate consequence of the fact that h (t) = t −1 I (tu), tu .
From Lemma 3.1, we can get the following lemma easily.
Then the functional I satisfies the following mountain pass geometry: Proof. From Lemma 3.2, we can get the existence of the (C) c sequence, we only need to prove {v n } is bounded. First of all, we observe that if a sequence {v n } ⊂ E satisfies for some constant C 1 > 0, then the sequence {v n } is bounded in E. For that, we simply need to demonstrate that R N v 2 n dx is bounded. In fact, by Lemma 2.1-(9) and (V ), we observe that Moreover, by the Sobolev inequality and Lemma 2.1-(9), one deduces Hence there is a constant C 3 > 0 such that Therefore, it remains to show that Let {v n } ⊂ E be an arbitrary Cerami sequence for I at level c > 0, that is and for any ϕ ∈ E, from Lemma 2.1- (6), we get ϕ n 2 ≤ 2 v n 2 and By computing (16) − 1 4 (17), one gets Thanks to (5), we get 1 Denote w n = f (v n ), then |∇v n | 2 = (1 + 2w 2 n )|∇w n | 2 . We can rewrite (16), (18) as follows.
GROUND STATE SOLUTIONS FOR QUASILINEAR SCHRÖDINGER EQUATIONS 1129 (20) From (20) and (V ), (K), we can see that {w n } is bounded in E, and there is C 5 > 0 such that It follows from (10), (21) and (K) that By the above inequality and (19), one has This completes the proof. Lemma 3.5. Assume that (V ), (g 1 ), (g 2 ) and (1) of (g 4 ) hold. If {u n } is bounded in E and u n → 0 in L α loc (R N ) for α ∈ [2, 2 * ), one has Proof. (i)The proof of (22). Firstly, when k(x) ∈ F 0 , we claim that for any > 0, there exists R > 0 such that where C 0 , C 1 are positive constants and independent on . (24) has already been proofed in [17], we sketch the proof as following. By the definition of F 0 , for any > 0, there exists R > 0 such that Let Ω i = {x ∈ B 1 (y i ) : |k(x)| ≥ }, then |Ω i | < , for all |y i | ≥ R . Now, covering R N by balls B 1 (y i ), i ∈ N, in such a way that each point of R N is contained in at most N + 1 balls. Without loss of generality, we suppose that |y i | < R , i = 1, 2, · · · , n and |y i | ≥ R , i = n + 1, n + 2, · · · , +∞. By the Hölder and Sobolev inequalities, we have Ωi (3) and (24), one has Let → 0, (22) is proved.
Covering R N by balls B 1 (y i ), i ∈ N, in such a way that each point of R N is contained in at most N + 1 balls. Without loss of generality, we suppose that |y i | < R , i = 1, 2, · · · , n and |y i | ≥ R , i = n + 1, n + 2, · · · , +∞. By the mean value theorem, there exists t n ∈ [0, 1] such that Set Then we have It follows from (8) and Lemma 2.1-(3) (7) that

GROUND STATE SOLUTIONS FOR QUASILINEAR SCHRÖDINGER EQUATIONS 1135
Hence we obtain Let → 0 and then δ → 0, we complete the proof of (26).
As the argument in [35](p.73, Theorem 4.2), we could obtain the following lemma which is also valid for functional I.
Next, we do some estimates and the method comes from the pioneering work [2] due to Brezis and Nirenberg. Without loss of generality, we assume that x 0 given by the condition (K) is the origin of R N and that B 2 (0) ⊂ Ω given by the condition (g 5 ).
Given > 0, we consider the function w : R N → R defined by . We observe that {w } is a family of functions on which the infimum that defines the best constant S for the Sobolev embedding D 1,2 (R N ) → L 2 * (R N ), is attained.
Then there exist positive constants k 1 , k 2 and 0 such that {x∈R N :|x|≤1} As → 0, we have
The proof of this lemma can be found in [33], we omit it.

GROUND STATE SOLUTIONS FOR QUASILINEAR SCHRÖDINGER EQUATIONS 1139
Arguing as in the case N = 3, we could also get Then we can get the conclusion by choosing A 0 large enough.
We also can get the conclusion by choosing A 0 large enough.
Case (4): N ≥ 10. Invoking (32) and (33), we get By (ii) of (g 5 ), (39) and Lemma 2.1-(9), one has for x ∈ B (0) and 0 < < 1 . Hence, similar to the proof of (42), we have where 0 < < 1 . Consequently, one gets Let A 0 large enough, we obtain Remark 2. From Lemma 3.9, we can get that c < 1 Proof of Theorem 1.1. Firstly, we invoke Lemma 3.2 to find a (C) sequence on level c, that is {u n } ⊂ E such that I(u n ) → c and (1 + u n ) I (u n ) → 0 with c = inf u∈N I(u). Applying Lemma 3.3, we may get, up to a subsequence, u n u in E, u n → u in L 2 loc (R N ) and u n (x) → u(x) a.e. in R N . For any φ ∈ C ∞ 0 (R N ), one has 0 = I (u n ), φ + o n (1) = I (u), φ , i.e. u is a weak solution of problem (3). The proof of this result is standard, so we omit it here.
By (45), (48) and the Sobolev inequality, one has which is a contradiction with Remark 2, thus β > 0. By the definition of β, there exist R > 0 and z n ∈ Z N such that If z n is bounded, there exists R > 0 such that which is a contradiction with u n → u = 0 in L 2 loc (R N ). Thus z n is unbounded, going if necessary to a subsequence, |z n | → ∞. Let w n (x) := u n (x + z n ), then there is a function w ∈ E such that w n w in E, w n → w in L 2 loc (R N ) and w n (x) → w(x) a.e. in R N .
On the one hand, by Lemma 3.1, for any u ∈ E, u = 0, there is a unique t u > 0 such that t u u ∈ N . Moreover, the maximum of I(tu) for t ≥ 0 is achieved at t u . Combining with V (x) ≤ V 0 (x) and G(x, s) ≥ G 0 (x, s), we obtain c ≤ I(t u u) ≤ I 0 (t u u) ≤ max t>0 I 0 (tu), hence c ≤ inf u∈E max t>0 I 0 (tu). It follows from Lemma 3.7 that c ≤ c 0 .
On the other hand, by (22), (23), V (x) ≤ V 0 (x), g(x, s) ≥ g 0 (x, s), Lemma 2.1-(8), (5), (44) and the Fatou lemma, we obtain c =I(u n ) − 1 2 I (u n ), u n + o n (1) G(x, f (u n ))dx + o n (1) Hence I 0 (w) = c 0 = c. Lemma 3.1 implies that there exists a unique t w > 0 such that t w w ∈ N . Then we have c ≤ I(t w w) ≤ I 0 (t w w) ≤ I 0 (w) = c, i.e. c is achieved by t w w. It follows from Lemma 3.4 that t w w is a ground state solution of problem (3). From (i), (ii), we can get that problem (3) has a nonnegative ground state solution u ∈ E. Furthermore, the stronger maximum principle implies u > 0, and Theorem 1.1 is proved.