EXISTENCE OF GROUND STATE SOLUTIONS FOR A CLASS OF QUASILINEAR SCHRÖDINGER EQUATIONS WITH GENERAL CRITICAL NONLINEARITY

In this paper, we study the following quasilinear Schrödinger equation −∆u + V (x)u−∆(u)u = g(u), x ∈ R , where N > 4, 2∗ = 2N N−2 , V : R N → R satisfies suitable assumptions. Unlike g ∈ C1(R,R), we only need to assume that g ∈ C(R,R). By using a change of variable, we obtain the existence of ground state solutions with general critical growth. Our results extend some known results.


1.
Introduction. This article is concerned with the following quasilinear Schrödinger equation −∆u + V (x)u − ∆(u 2 )u = g(u), x ∈ R N , (1.1) where N > 4 and 2 * = 2N N −2 . It is well known that it is a hot problem in nonlinear analysis to study the existence of solitary wave solutions for the following quasilinear Schrödinger equation where z : R × R N → C, W : R N → R is a given potential, l : R → R and k : R N × R → R are suitable functions. For various types of l, the quasilinear equation If we take g 2 (u) = 1 + [(l 2 (u)) ] 2 2 , then (1.3) turns into quasilinear elliptic equations (see [32]) − div(g 2 (u)∇u) + g(u)g (u)|∇u| 2 + V (x)u = h(x, u), x ∈ R N . (1.4) For (1.4), there are many papers (see [32,14,13,4,5]) studying the existence of positive solutions. But in our mind, there are few papers to study the existence of ground state solutions for this problem. If we set g(s) = √ 1 + 2s 2 and h(x, u) = h(u) in (1.4), then (1.4) reduces to the Equ. (1.1). In [31], Poppenberge-Schmitt-Wang consider an eigenvalue problem for the following equation where V (x) ∈ L 1 loc (R N ), inf x∈R N V (x) > 0 and κ > 0 is a parameter. By using constrained variational method, they prove the existence of a positive solution of problem (1.5) in H 1 (R N ) for certain κ > 0 with some further conditions on V (x) and N = 1. Moreover, for p ≥ 3, a positive solution was established for any κ > 0 by the Mountain Pass Theorem. If N ≥ 2 and 1 < p < 2 * − 1, in [31], the authors obtain the existence of a nontrivial nonnegative solution in H 1 r (R N ) for certain κ > 0. Furthermore, for N = 3 and 3 ≤ p < 5, the authors [31] show that problem (1.5) has a nontrivial nonnegative solution in H 1 r (R N ) for all κ > 0. Based on the work in [31], many papers focused on subcritical case, see [24,23,29,30]. As the argument in [22], 22 * behaves like a critical exponent for (1.5), and they gave an open question on whether there existence results for p = 22 * . For this open question, many papers studied the existence of solutions with critical exponent 22 * . For more specific works, please see [15,25,26]. In particular, in [15], doÓ et al. considered the following quasilinear Schrödinger equations − ∆u + V (x)u − ∆(u 2 )u = |u| q−1 u + |u| p−1 u in R N , (1.6) where 3 < q < p ≤ 22 * − 1 and 2 * = 2N N −2 is the critical exponent. They also assume that V (x) satisfying the following conditions (V 1 ) the function V : R N → R is continuous and uniformly positive, that is, there exists a constant V 0 > 0 such that where the last inequality is strict on subset of positive measure in R N ; (V 2 ) the function V is periodic in each variable of x 1 , . . . , x N . For 3 < q < 22 * − 1 and p = 22 * − 1, a positive classic solution of (1.6) in H 1 (R N ) with (V 1 )-(V 2 ) was obtained in [15] by introducing the changing of variable v = f −1 (u) (see [22,9,16]) and transforming (1.6) into a semilinear equation. For more semilinear problems, we refer to [35,39,8] and so on. Following this work, in [33], Silva and Vieira studied the following quasilinear Schrödinger equation By assuming that V (x), K(x) and g(x, u) are asymptotically periodic and satisfy some further conditions, the authors [33] established the existence of nontrivial solution by the concentration-compactness principle and a comparison argument. Specifically speaking, the authors used a change of variable to reformulate the problem obtaining a semilinear problem which has an associated functional well-defined in H 1 (R N ) and satisfies the geometric hypotheses of the Mountain Pass Theorem. They considered the functional associated with the modified problem and used a version of the Mountain Pass Theorem without compactness condition to get a Cerami sequence associated with the minimax level. Moreover, this sequence and a technical results are applied to get a nontrivial critical point of the functional associated with the periodic problem. Furthermore, there are many papers forcing on the existence and concentration of positive ground and bound state solutions for (1.1) with critical growth. For the results of this type, we refer to [18,36,38].
Very recently, in [6] the first author, Tang and Cheng considered the problem (1.1) with constant potential and subcritical nonlinearity, which was proved by the critical point theorem developed by [19]. Based on this work, the first author, Tang and Cheng [7] considered the problem (1.1) with critical nonlinearity and established the existence of positive ground state solutions via Pohažaev manifold.
It is a natural problem if the existence of positive ground state of (1.1) with general critical nonlinearity? To our knowledge, the existence of the ground solutions to Eq. (1.1) with general critical nonlinearity term has not ever been considered by variational methods. In this paper, we will give an affirmative answer for the above question. The underling idea for proving our results is motivated by the method used in [11,27,1,34]. As before mentioned papers, we also use a change of variable to reformulate the problem obtaining a semilinear equation. It is worth pointing out that there are many difficulties in treating this class of quasilinear Schrödinger equation in R N . To this end, we need to overcome those difficulties: (I) it is the lack of compactness; (II) the estimate of mountain pass level. Now, let us recall some basic notions. Let and we denote by L p (R N ) the usual Lebesgue space with norms Next, we consider the problem (1.1) with general critical nonlinearity. Before stating our results, we need to give some certain conditions on g.
Secondly, we give the second result in this paper. Before stating our result, assume that g satisfied some certain conditions (g 1 )-(g 3 ) and V (x) is not a constant.
. Now, we are ready to state the second result of this case as follows.
. They obtained positive ground state solutions when g satisfies (g 1 )-(g 3 ) and , for t > 0 and some C > 0.
But in the present paper, (g 4 ) is removed.
). An example is given by: where N > 4, C N is the surface area of the N -dimensional unit ball and S * is the best constant of the embedding D 1,2 (R N ) → L 2 * (R N ) and V ∞ > 2 is a positive constant.
The remainder of this paper is organized as follows. In section 2, we list some useful propositions and lemmas, which play an important role in proving our results. The proofs of Theorem 1.1 and Theorem 1.2 are given in Section 3 and Section 4, respectively.
In this paper, R N ♣ denotes R N ♣ dx and C denotes the different constants.

2.
Variational framework and some preliminary lemmas. In this section, we give some useful lemmas. Let endowed with the norm By the conditions (V 1 ) and (V 2 ), it is easy to check that · E is equivalent to the norm · . In (1.1), we can deduce formally that the Euler-Lagrange functional associated with the equation (1.1) is For (1.1), due to the appearance of the nonlocal term R N u 2 |∇u| 2 , J may be not well defined. To overcome this difficulty, we apply an argument developed by Liu et al. [22] and Colin and Jeanjean [9]. We make the change of variables by v = f −1 (u), where f is defined by It is easy to check to that I ∈ C 1 . We also know that if v is a critical point of the functional I, then u = f (v) is a critical point of the functional I, i.e. u = f (v) is a solution of problem (1.1).
Next, let us recall some properties of the change of variables f : R → R, which are proved in [22,9,16] as follows: 22,9,16]). The function f (t) and its derivative satisfy the following properties: (1) f is uniquely defined, C ∞ and invertible; for all t ∈ R; (8) |f (t)| ≤ 2 1/4 |t| 1/2 for all t ∈ R; (9) there exists a positive constant C such that Next, we need to recall a critical point theory, which is given as follows: 19]). Let (X, · ) be a Banach space and I ⊂ R + an interval. Consider the following family of C 1 -functionals on X: We assume there are two points v 1 , v 2 in X such that Then for almost every λ ∈ I there is a sequence {v n } ⊂ X such that Moreover, the map λ → c λ is non-increasing and continuous from the left. |v n | r = 0, Lemma 2.4 (see Lemma 8.9 in [37]).
Next, we introduce the following lemma, which was proved in [3]. Let {v n }, v and w be measurable function from R N to R, with w bounded, such that By the Brezis-Lieb Lemma in [37], we can prove the follow lemma.
Lemma 2.6 (see Lemma 2.1 in [41]). Assume that h ∈ C(R N × R) and there exists a constant C 0 > 0 such that where H(x, s) = s 0 h(x, t)dt. By a standard argument in [37], we can obtain the following Pohozaev type identity.
3. Proof of Theorem 1.1. In this section, we want to give a proof of Theorem 1.1. At first, if we choose V (x) ≡ 1 in Lemma 2.7, then we can get the following version of Pohozaev identity.
We define the following energy functional and By Sobolev inequality and Lemma 2.1-(9), we can get In fact, for any v ∈ H 1 (R N ), by Sobolve ineqaulty and Lemma 1.1, we have that
(iii) For Next, we shall estimate c λ . Let η ∈ C ∞ 0 (R N , [0, 1]) be a cut-off function such that η ≡ 1 on B 1 (0) and η ≡ 0 on R N \B 2 (0), where B 1 (0) denotes the ball in R N of center at origin and radius 1. Given ε > 0, we consider the function U ε : We know that {U ε } ε>0 is a family of functions on which the infimum, that defines the best constant, S, for the embedding D 1,2 → L 2 * (R N ), is attained. Let us define The following lemmas were proved in [18,36], we just state them as follows. 18,36]). If ε → 0, then v ε satisfies the following useful estimates: By the mountain pass value, one has By Lemma 3.3, we know that v ε ∈ X. Now, by Lemma 3.3 and (g 3 ), we have that On the one hand, from which implies that t ε is bounded from above by some T 1 > 0. On the other hand, by (3.12), one has Letting ε → 0, by (3.9), (3.10), (3.11) and 4 < q < 22 * , we have That is, we get a lower bound for t ε independent of ε. Next, we estimate H λ (t). Note that the function for ε > 0 small enough. This completes the proof.

EXISTENCE OF GROUND STATE SOLUTIONS 503
Choosing ε > 0 sufficiently small and by using v λ = 0, we know that there exists > 0 such that v λ ≥ > 0.
Let v 1 n = v n − v 0 . Thus by Brezis-Lieb Lemma and Lemma 2.6, we have that the following hold:

It is a standard to prove (i)-(iii) via Brezis-Lieb Lemma. Next, if we set
then we write Eq. (1.1) in the following form Similar to the proof of Lemma 2.2 of [12], it is easy to check that (3.15) By (3.15) and Lemma 2.6, we have Thus it follows from v 1 n 0 and (iii) that which implies that (iv) holds. Next, we prove (v). By (3.15), for any ε > 0, there exists a constant C ε > 0 such that |h λ (v n )| ≤ ε(|v n | + |v n | 2 * −1 ) + C ε |v n | r , for all 2 < r < 2 * .
. Then there exists a subsequence of {v n }, still denote by {v n }, such that Proof. By the boundedness of {v n }, there exists v λ ∈ H 1 (R N ) such that v n v λ in H 1 (R N ). From Remark 3.5, we assume that v n ≥ 0 in H 1 (R N ). If v λ = 0, then the have finished. Otherwise, we may suppose that v n 0 in H 1 (R N ). Next, we will show that there exists δ > 0 such that lim n→∞ sup y∈R N B1(y) |v n | 2 ≥ δ > 0. (3.25) If (3.25) does not hold, then by Lemma 2.3, we know that v n → 0 in L r (R N ) for any r ∈ (2, 2 * ). Similar to (3.22), we have By (3.13) and Φ λ (v n ) → c λ and Φ λ (v n ) → 0, one has (3.27) Assume that v n → σ for some σ > 0, by Sobolev embedding, then we know that , which together with (3.26) and (3.27), it follows that This is a contradiction. Thus we can infer that (3.25) holds. Thus there exists a sequence {y n } ⊂ R N such that |y n | → +∞ and Set w n = v n (· + y n ). Thus we get Φ λ (w n ) → c λ and Φ λ (w n ) → 0. By (3.25), we deduce that w n w λ = 0 in H 1 (R N ) and Φ λ (w λ ) = 0. It is easy to check that w λ ≥ 0 in R N and due to the fact w λ = 0, we get w λ > 0 in R N .
Proof of Theorem 1.1. By Lemma 3.8, for a.e. λ ∈ [1/2, 1 and Φ λ (v n ) → 0. Using Lemma 3.7, we have that Next, by the Sobolev embedding inequality, we have that R N |∇v λn | 2 ≤ C and v λn 2 * ≤ C for all n ∈ N. Combine (g 1 )-(g 3 ), Lemma 2.1-(8) with Lemma 3.1, we deduce that for any ε > 0, there exists C ε > 0 such that Therefore, choosing ε < 1/2, we can infer that there exists C > 0 such that By Sobolev inequality and Lemma 2.1-(9), we can get (3.30) Thus we have that {v λn } is bounded in H 1 (R N ). Next, we can assume that the limit of Φ λn (v λn ) exists. By Theorem 2.2, we know that λ → c λ is continuous from the left. Thus we get Then by using the fact that and lim n→∞ Φ (v λn ) = 0.
By Remark 3.6, there exists > 0 independent of λ n such that v λn ≥ . Similar to the proof of Lemma 3.8, we can obtain the existence of a positive solution v 0 for (1.1). By Lemma 3.7, we have Next, by the definition of m, we can find {ν n } ⊂ H 1 (R N ) such that Φ(ν n ) → m and Φ (ν n ) → 0. From Remark 3.6, we deduce that ν n ≥ , where > 0 is independent of n. Similar to the proof of (3.30), we deduce that {ν n } is bounded in H 1 (R N ). In virtue of Remark 3.5, we may assume that ν n ≥ 0 in H 1 (R N ). Taking in mind that ν n ≥ > 0, we can proceed as in proof of Lemma 3.8, to show that there exists {ν n } ⊂ H 1 (R N ) such that ν n ≥ 0, ν n ν 0 > 0 in H 1 (R N ), Φ(ν n ) → m and Φ (ν n ) → 0. By Lemma 3.7, we have that Φ(ν 0 ) ≤ m and Φ (ν 0 ) = 0. In order to complete the proof, we need to prove that Φ(ν 0 ) ≥ m. In fact, since Φ (ν 0 ) = 0, we also get Φ(ν 0 ) ≥ m. Thus we have that Φ(ν 0 ) = m and Φ (ν 0 ) = 0. The proof is completed.
4. Proof of Theorem 1.2. In this section, we want to prove Theorem 1.2, that is, we prove the existence of ground state solution for (1.1) with the assumptions that V (x) is not equality to a constant. For this case, we will assume that V (x) ≡ V ∞ . From (1.6), for λ ∈ [1/2, 1], we introduce the following family of functionals by where v ∈ H 1 (R N ).
Lemma 4.1. Let v ∈ E be a nontrivial critical point for Then there exists γ ∈ C([0, 1], By a directly calculation, it is easy to see that v t has the following properties: and we can see that Hence we can deduce that max < 0 and set γ(t) = v αt for t ∈ (0, 1] and γ(0) = 0, we obtain the desired γ. This completes the proof.
and Ψ λ (v n ) → 0, then there exists a subsequence {v n }, Proof. Since {v n } is bounded in H 1 (R N ), we can assume that v n v λ . By the proof of Lemma 3.7 and using (V 2 ), we can easily prove that Ψ λ (v λ ) = 0 and Ψ λ (v λ ) → c λ . Thus (i) holds.