GROUND STATES FOR KIRCHHOFF-TYPE EQUATIONS WITH CRITICAL GROWTH

. In this paper, we study the following Kirchhoﬀ-type equation with critical growth where a > 0, b > 0, λ > 0 and f is a continuous superlinear but subcritical nonlinearity. When V and f are asymptotically periodic in x , we prove that the equation has a ground state solution for large λ by Nehari method. Moreover, we regard b as a parameter and obtain a convergence property of the ground state solution as b (cid:38) 0.


QUANQING LI, KAIMIN TENG AND XIAN WU
which is related to the stationary analogue of the Kirchhoff equation which was proposed by Kirchhoff [15] as an extension of the classical D'Alembert wave equation for free vibrations of elastic strings. In (1.2), u denotes the displacement, g(x, u) the external force and b the initial tension while a is related to the intrinsic properties of the string, such as Young' modulus. We have to point out that such nonlocal problems also appear in other fields as biological systems, where u describes a process which depends on the average of itself, for example, population density. After the pioneer work of Lions [16], in which a functional analysis approach was introduced, (1.2) has been paid much attention to by many researchers. Some early studies of the Kirchhoff equation may refer [2,3,5,7,9,16,20]. Recently, some scholars have studied the existence of nontrivial solutions for the Kirchhofftype problem where a > 0, b ≥ 0, N = 1, 2, 3, V : R N → R and g ∈ C(R N × R, R). For example, Wu in [23] studied the sequence of high energy solutions for Eq. (1.3) and four main results were given. By some special techniques, the authors in [18] improved and united the above four results. In [6], problem (1.3) with concave and convex nonlinearities was studied by using Nehari manifold and fiber map methods, and multiple positive solutions were obtained. Li and Ye in [19] proved that Eq. (1.3) with g(x, u) = |u| p−1 u had a ground state solution, where 2 < p ≤ 3. For more results, we refer the readers to the papers [8,10,11,12,21,24] and the references therein.
In this paper, we shall study the existence of ground states for critical problem (1.1).
Let H be the class of functions h ∈ L ∞ (R 3 ) such that, for every ε > 0 the set {x ∈ R 3 : |h(x)| ≥ ε} has finite Lebesgue measure. In order to reduce the statements for main results, we list the assumptions as follows: and there exists 4 < q < 6 such that where h ∈ H and q is given by (f 1 ).
is the function f (x, t) = |t| q−2 t and we may choose . We point out that this kind of hypothesis (f 3 ) was first introduced by Jeanjean in [14]. Latter, it was used by Liu and Li [17] for the general case. Furthermore, Remark 1.1 in [25] implies that (f 3 ) is weaker than the following assumption: is non-increasing on (−∞, 0) and non-decreasing on (0, +∞). Set and the norm u = u, u 1/2 .
In view of (V 1 ) − (V 2 ), the norms u and u p : 1 2 are equivalent. By the condition (V 1 ) we know that the embedding H 1 V (R 3 ) → L s (R 3 ) is continuous for each 2 ≤ s ≤ 6 and locally compact for each 2 ≤ s < 6.
Our main results are the following: Then there exists λ * > 0 such that for each λ > λ * , problem (1.1) has a ground state solution u λ .
Remark 1. When f is a function of C 1 class, He and Zou in [13] the authors in [1] proved that there exists λ * > 0 such that the above equation has a positive solution for all λ ≥ λ * , where V is asymptotically periodic and M : R + → R + satisfies the following assumptions (M 1 ) There exists a > 0 such that M (t) ≥ a, ∀t > 0. (M 2 ) There exists b ≥ 0 such that lim Then H(b) = 1 and the above problem reduces to the problem but at this stage, the condition (f 3 ) in the present article is weaker than (f 5 ) * .
Fixed λ > λ * . Obviously, u λ obtained in Theorem 1.1 depends on b, we next denote u λ by u b λ to emphasize this dependence. As like [8,21], we give a convergence property of u b λ as b 0, which reflects some relationship between b > 0 and b = 0 in problem (1.1). Our main result in this direction can be stated as the following theorem.
Theorem 1.2. If the assumptions of Theorem 1.1 are verified, then, for any sequence {b n } with b n 0 as n → ∞, there exists λ * > 0, independent of b n , such that for each λ > λ * , problem (1.1) has a ground state solution u bn λ with u bn It is well known to us that a weak solution of problem (1.1) is a critical point of the following functional 2. Proof of Theorem 1.1. To begin with, we give some lemmas.
for small t > 0, and (2.1) implies that for small t > 0. Moreover, by virtue of (f 4 ) we have as t → +∞. Hence α has a positive maximum and there exists t u > 0 such that α (t u ) = 0 and α (t) > 0 for 0 < t < t u . We claim that α (t) = 0 for all t > t u . Indeed, if the conclusion is false, then, from the above arguments, there exists t u < t 2 < +∞ such that α (t 2 ) = 0 and Combining the claim with prior arguments, we obtain the first conclusion of (i). The second conclusion is an immediate consequence of the fact that α (t) = t −1 I λ (tu), tu . This completes the proof of (i). Similarly, we can prove that (ii) holds.
(iii) For u ∈ S 1 , by (i) there exists t u > 0 such that t u u ∈ N . Hence for ε > 0 small, (2.1) implies that As a consequence, there exists t 0 > 0 such that t u ≥ t 0 for all u ∈ S 1 . To prove that t u ≤ C W for all u ∈ W ⊂ S 1 . We argue by contradiction. Suppose there exists {u n } ⊂ W ⊂ S 1 such that t n := t un → +∞ as n → ∞. Since W is compact, there exists u ∈ W such that u n → u in H 1 V (R 3 ). Consequently, by (f 4 ) we deduce that |u n | 6 dx ≥ 0, a contradiction. So the conclusion (iii) follows.
(iv) For λ > 0 and u ∈ S ρ , by the proof of (i) we know that for small ρ > 0. Furthermore, for every u ∈ N , there exists t 1 > 0 such that t 1 u ∈ S ρ . Hence by (i) we have This completes the proof.
In view of Lemma 3.8 in [4] we know that w n → 0 in L 6 (R 3 ). By the interpolation inequality, one has where s ∈ (2, 6) and θ = 3(s−2) 2s . Then w n → 0 in L s (R 3 ) for 2 < s < 6. Hence by (2.2), for large t and small ε > 0, we have as n → ∞, a contradiction. Consequently, by (f 4 ) we obtain for large n, a contradiction. This completes the proof.
Remark 3. If we regard b > 0 in problem (1.1) as a parameter and denote c λ by c b λ , then from the above proof we see that c b λ → 0 as λ → +∞ uniformly in b ∈ (0, 1]. Hence λ * in Lemma 2.3 is independent on b ∈ (0, 1]. This will be used to prove Theorem 1.2 in section 3. Define the mapping m : S 1 → N by setting m(w) := t w w, where t w is as in Lemma 2.1.

Lemma 2.4 ([22]
). The mapping m is a homeomorphism between S 1 and N , and the inverse of m is given by m −1 (u) = u u .
Proof of Theorem 1.1. By Lemma 2.5 (iii), it suffices to prove that the infimum c λ is attained for fixed λ > λ * . For fixed λ > λ * , let {w n } ⊂ S 1 be a minimizing sequence satisfying ψ λ (w n ) → c λ = inf S1 ψ λ by Lemma 2.5 (ii). By the Ekeland variational principle, we suppose . Set u n = m(w n ) ∈ N . Then Lemma 2.5 (i) implies that I λ (u n ) = ψ λ (w n ) → c λ and I λ (u n ) → 0 in H −1 V (R 3 ). By Lemma 2.2 we see that {u n } is bounded in H 1 V (R 3 ). Therefore, up to a subsequence, there exists u ∈ H 1 V (R 3 ) such that u n u in H 1 V (R 3 ), u n → u in L s loc (R 3 ) for 2 ≤ s < 6 and u n (x) → u(x) a.e. on R 3 . Hence I λ (u) = 0. In the following, we distinguish two cases.
Case 1: u = 0. Then u ∈ N and c λ ≤ I λ (u). Consequently, by weakly lower semi-continuity of the norm, Fatou Lemma and (f 3 ) one has which implies that I λ (u) = c λ . Case 2: u = 0. We shall apply the concentration-compactness principle due to P. L. Lions to the sequence of L 1 functions u 2 n and we know that two cases may happen.
Step 2: We claim that I λ,p (ū) ≤ c λ . Indeed, by the boundedness of { u n } and Lemma 2.6 we have Again by the periodicity of V p and f p in the variable x, weakly lower semi-continuity of the norm, (f 5 ) − (iii) and Fatou Lemma we deduce that