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

. We establish the existence of ground state solution for quasilinear Schr¨odinger equations involving critical growth. The method used here is min- imizing the gradient integral norm in a manifold deﬁned by integrals involving the primitive of the nonlinearity function.

1. Introduction. The study of the minimization problem min{ 1 2 R N |∇u| 2 dx : R N G(u)dx = 1}, for N ≥ 3, min{ 1 2 R N |∇u| 2 dx : R N G(u)dx = 0}, for N = 2, has a great importance for the existence of solutions for the equation where g(u) = G (u) and G ∈ C 1 (R, R). In [4], for N ≥ 3, and [3], for N = 2, the authors studied the problem (1) in order to establish the existence of ground state solution for (2) provided that g ∈ C(R, R) is odd and satisfies for all s ≥ 0.
In [2], the authors completed this study of (2) for a class of nonlinearities with critical growth. The aim of the present paper is to extend the method which has been used in [2] to obtain a ground state solution to the quasilinear Schrödinger equation where N ≥ 2 and we assume that h : R → R is a continuous function which is odd and satisfies: (h 4 ) There is λ > 0 and q ∈ (4, 2(2 * )) when N ≥ 3, or q > 4 when N = 2, such that h(s) ≥ λs q−1 , for every s ≥ 0.
The condition (h 2 ) says that h has a critical growth at infinity. As observed in [14] (see also [19]), the number 2(2 * ) behaves like a critical exponent for equation (3) when N ≥ 3, while the exponential growth above is the critical growth for this kind of problem when N = 2. According to [10,11], the critical growth when N = 2 is the following: Note that (h 2 ) in the case N = 2 is a slightly weakened version of (CG) and says that h has a critical exponential growth at infinity. Consequently, the so-called Trudinger-Moser inequality [6,17,20] plays a crucial role in our arguments. We observe that the condition (h 4 ) implies that (g 3 ) holds. It is worth pointing out that (g 3 ) is a necessary condition for the existence of a solution of (4), as we can see using the Pohozaev's identity (see [4,Proposition 1]). The quasilinear equation (3) has a great relevance because it appears in mathematical models associated with several physical phenomena (see [5,9] or [18] for a complete bibliography about other contexts of physics where quasilinear Schrödinger equations arise).
Formally, the problem (3) is the Euler-Lagrange equation for the energy functional Because of the term u 2 |∇u| 2 , the natural function space associated with the functional J is given by A function u ∈ S is called a weak solution of (3) if . It is not difficult to verify that the derivative of J in the direction φ at u is Thus, u ∈ S is a weak solution of (3) if, and only if, the derivative J (u)φ = 0 for every direction φ ∈ C ∞ 0 (R N ). As defined in [8], we say that a weak solution u of (3) is a ground state if J(u) = inf{J(w) : w is a nontrivial weak solution of (3)}.
In order to obtain a ground state solution of (3), we proceed as in [9,14] by changing of variables v = f −1 (u), where f is defined by The transformation f is the key in many arguments here, it was used first independently in [9] and [14]. Thus, we can write J(u) as In [9], it is proved that that I is well defined on the usual Sobolev space H 1 (R N ) and then Actually, we prove in the Appendix the following proposition, which will be useful to study the equation (3). Our aim is to show the following results: Theorem 1.1 provides a ground state solution of (4), that is, a nontrivial solution w ∈ H 1 (R N ) of (4) such that I(w) ≤ I(v) for every nontrivial solution v of (4). As a consequence we have the following result: Corollary 1. Under the hypotheses of Theorem 1.1, then problem (3) (respectively, the dual problem (4)) possesses a nontrivial ground state solution.
The results of Theorem 1.1 and Corollary 1 also hold for N = 2, more precisely:   Positive and sign-changing ground state solutions were considered in [15,16] for a class of quasilinear Schrödinger equations involving several types of potentials; however, the corresponding problems have been studied for the subcritical case with nonlinearities requiring the assumption h(s)/s is increasing in (0, ∞) and the Ambrosetti-Rabinowitz condition.
The paper is organized as follows. The section 2 is dedicated to fix some notations. The case N ≥ 3 is treated in Section 3, while the case N = 2 is considered in Section 4. In the Appendix, we prove Proposition 1.

2.
Notation. We consider the functional I : H 1 (R N ) → R associated with (4) and defined by and H(s) = s 0 h(τ )dτ . Since f and h are odd functions, the function G is even. Set m = inf{I(v) : v is nontrivial solution of (4)} (5) and while We also define the following minimax value Observing that the problem (4) is autonomous, the Schwarz symmetrization can be used to reduce the minimizing problem (6) in the space H 1 rad (R N ) of the radial functions.
The main feature of the proof of our results is the verification that A is attained and m = A = b, when N = 2, or In the remainder of this section we gather some properties satisfied by the function f . These were proved first independently in [9] and [14] (see also [1]).
3. The case N ≥ 3. Throughout this section, S denotes the best constant of Sobolev embedding The proof of the following result is based on arguments found in [19].
Proof. We have From (9) and (10), it follows that Therefore, R N |v n | 2 dx is bounded and finally that (v n ) is bounded in H 1 (R N ). For simplicity of notation, we write We have The conditions ( Thus, Ψ (v) = 0 for every v ∈ M.
Remark 2. This last argument works when N = 2. In this case and the last inequality is strict.
The proof of the identity is based on arguments that will be used in the next section for the case N = 2, following Jeanjean and Tanaka [12].
From (9), As in the proof of Lemma 3.1, On the other hand, Taking n → ∞ in the above expression, we obtain a contradiction. Thus A > 0.
Since M is a C 1 manifold, from the Ekeland variational principle, there is v n ∈ M and λ n ∈ R such that Lemma 3.4. The sequence (λ n ) given by (13) is bounded and lim sup n→∞ λ n ≤ 2A.
Since R N |∇v n | 2 dx → 2A, (14) shows that lim sup n→∞ λ n ≤ 2A. As implies that λ n does not converges to 0. The concentration-compactness principle [13] applied to the sequence (v n ) guarantees the existence of positive finite measures µ and ν and nonnegative sequences ( where v is the weak limit of v n in H 1 (R N ).
Combining (iv) with Lemma 3.5, we deduce that there exists at most a finite number of points x i on bounded subsets of R N . Hence, given i, there exists δ i > 0 such that x j does not belong to the ball B(x i , δ i ), except for j = i. Let 0 < < δ i and ϕ ∈ C ∞ 0 (R N ), 0 ≤ ϕ(x) ≤ 1, ϕ ≡ 1 on B(x i , /2) and ϕ ≡ 0 on R N \ B(x i , ). By construction, we have Taking n → ∞ in the above inequality and taking → 0 in the resultant expression, we obtain . Thus, Let t λ > 0 be the global maximum point of the function We can check that and from (f 1 )-(f 6 ), we have that implies On the other hand, since from Remark 1 and (f 3 ), we can see that t λ → 0 as λ → ∞. To complete the proof, fix Λ > 0 such that , for all λ > Λ.
We observe that the choice of Λ depends only on q, ψ and its norms in L q (R N ) and H 1 (R N ).
Remark 4. As a consequence of Remark 3 and Lemma 3.7, the cases ρ > 0 and ν i > 0, for some index i, cannot coexist.
Lemma 3.8. The weak limit v of the minimizing sequence (v n ) is nontrivial.
Proof. Suppose, by contradiction, that u = 0. Consider ρ ≥ 0 given by (h 2 ). If We now assume ρ > 0. Since v n ∈ H 1 rad (R N ), v n (x) are uniformly bounded in |x| ≥ δ, for every δ > 0. As a consequence, every number ν i is null, except eventually the corresponding number associated with the atom at the origin. Let ν 0 denote this unique number which can be assumed nonnegative. Invoking Remark 4, we get ν 0 = 0, and we conclude that (19) holds. Arguing as in the proof of Lemma 3.3, we obtain the same contradiction. Therefore, v = 0. Lemma 3.9. The constant A is attained by the weak limit v of v n .
In case ρ = 0, Lemma 3.5 implies that On the other hand, if ρ > 0 then Lemmas 3.2, 3.6 and 3.7 imply that ν i = 0 for every i. Thus, we obtain that (20) holds. Recall that We claim that which gives In effect, set It is easy to check that (h 1 ) and (h 2 ) imply that By Fatou lemma, after passing to the limit in above equality, we find and the claim (21) holds. We claim now that v ∈ M. Indeed, if v ∈ M, one would have which is a contradiction. Therefore, v ∈ M and T (v) = A. Theorem 1.1 is proved.
We are now ready to prove Corollary 1. In fact Corollary 1 may be proved in much the same way as [4, Theorem 3, p.331]. By Theorem Since M a C 1 manifold (see Remark 2), by Lagrange multipliers, there is θ ∈ R such that we have θ > 0. Setting v(x) = w x √ θ , we see that v is also a nontrivial solution of (4).

From (26) and (27)
On the other hand, combining (23), (24), (25), (26) and (27), we have thanks to (28). This implies that v is a ground state solution for problem (4). We claim that u = f (v) is a ground state solution for problem (3). Effectively, let φ ∈ S be any nontrivial solution of (3) and set w = f −1 (φ). According to [8] (see also [1]), J(φ) = J(f (w)) = I(w). From the properties of f and Proposition 1, w is a nontrivial critical point of I and then I(w) ≥ I(v). Consequently, Hence, u is a ground state solution in S, which completes the proof of Corollary 1.

4.
The case N = 2. We begin by recalling that Pohožaev's identity (see [4, Proposition 1]) shows that R 2 G(v)dx = 0 is a necessary condition for the existence of a solution of (4). Throughout the section, we continue to write M for the set Define the following minimax value: The first result in this section shows a sufficient condition on a sequence {v n } to get a convergence like H(f (v n )) → H(f (v)) in L 1 (R 2 ).
Proof. Since (v n ) is bounded in H 1 (R 2 ) and is weakly convergent to v, we may assume that v n (x) → v(x) almost everywhere in R 2 . We can also assume that lim |x|→+∞ v n (x) = 0, uniformly in n, because (v n ) is a bounded sequence in H 1 rad (R 2 ). By a version of Trudinger-Moser inequality [6], given positive numbers m < 1 andM , there exists a constant C = C(m,M ) > 0 such that for every w ∈ H 1 (R 2 ) such that ∇w 2 L 2 ≤m < 1 and w L 2 ≤M .
As in the preceding section, we derive some results involving the levels A and c.
From (h 1 ) and (h 4 ), we conclude that φ(t) < 0 for t sufficiently small and φ(t) > 0 for t sufficiently large. Thus, exists t 0 > 0 such that φ(t 0 ) = 0, hence that t 0 v ∈ P and finally Therefore, A ≤ c and the lemma follows. Proof. It is clear that A ≥ 0. Assume by contradiction that A = 0 and let (w n ) be a sequence in H 1 (R 2 ) satisfying For each λ n > 0, the function v n (x) = w n ( x λn ) satisfies Since we choose λ 2 n = 1/ R 2 f (w n ) 2 dx. In this way, Consequently, We stress that we may consider the sequence (v n ) in the space H 1 rad (R 2 ). From Lemma 4.1 This gives, because On the other hand, from (h 4 ), there exists q > 4 and λ > 0 such that Using (f 5 ), there exists a positive constant C such that where we have used that q > 4. Therefore, R 2 |v n | 2 dx is bounded and finally that (v n ) is bounded in H 1 (R 2 ). Hence there exist v ∈ H 1 rad (R 2 ) and a subsequence, still denoted by (v n ), such that (v n ) is weakly convergent to v in H 1 rad (R 2 ). By the weak convergence, Thus, and hence that v is almost everywhere a constant function. Using that v(x) = 0 as |x| → +∞, we conclude that v = 0, which is a contradiction with (35). Therefore, A > 0 and the lemma follows.  Proof. Let (v n ) be a minimizing sequence in H 1 rad (R 2 ) for A, that is, Setting v √ θ (x) = v x √ θ , we see that v √ θ is also a solution of (4) and satisfies Thus, For each γ ∈ Γ one has γ([0, 1]) ∩ P = ∅ (see [12]). Hence, there exists t 0 ∈ (0, 1] such that γ(t 0 ) ∈ P, and consequently Thus, which implies A ≤ b. From (39) and (40), we find On the other hand, let w ∈ H 1 (R 2 ) be a nontrivial solution of (4). Using the construction made in [12], there exists a path γ w ∈ Γ such that w ∈ γ w ([0, 1]) and max t∈[0,1] I(γ(t)) = I(w). Consequently, b ≤ I(w). Therefore, Combining (41) with (42), we obtain m = A = b. Thus I(v √ θ ) = m and it completes the proof of Theorem 1.2. The same reasoning applied to the case N ≥ 3 shows that u = f (v) is a ground state solution of (3), which proves Corollary 2.
Moreover, φ ∈ S satisfies (43). It is readily seen that I (v).w = J (u).φ. Since C ∞ 0 (R N ) is dense in H 1 (R N ), we have proved that v is a critical point for the functional I. We now consider the case N = 2. Let u ∈ S be a weak solution of (3). As H 1 (R 2 ) ⊂ L r (R 2 ) for every r ∈ [2, +∞), we have u ∈ L 4 (R 2 ) and we can now proceed analogously to the proof of case N ≥ 3 from (45) to conclude that v = f −1 (u) is a critical point of I. To prove the sufficient part of the proposition and the identity J(u) = I(v), we refer the reader to [9].