Ground State Solutions for a Quasilinear Kirchhoff Type Equation

We study the ground state solutions of the following quasilinear Kirchhoff type equation


Introduction and main results
Consider the following Kirchhoff type equation where b ≥ 0, 1 < p < 11, µ > 0 is a parameter and the potential V(x) satisfies the following condition: (V) V ∈ C(R 3 , R) satisfies inf V(x) = V 0 > 0 and for each M > 0, meas{x ∈ R 3 : V(x) ≤ M} < +∞, where meas denotes the Lebesgue measure in R 3 .
Problem (1.1) arises in an interesting physical context.In fact, if V(x) = 0 and replacing R 3 by a bounded domain Ω ⊂ R 3 in (1.1), problem (1.1) without the term [∆(u 2 )]u reduces to the following Dirichlet problem of Kirchhoff type which is related to the stationary analogue of the equation presented by Kirchhoff in [7], where ρ, ρ 0 , h, E and L are constants.Problem (1.3) extends the classical D'Alembert's wave equation, by considering the effects of the changes in the length of the strings during the vibrations.After Lions [10] proposed an abstract framework to the problem (1.2), many papers devoted to the existence of multiple nontrivial solutions, infinitely many solutions and ground solution to the semiliner (without the term [∆(u 2 )]u) Kirchhoff type problems by applying the modern variational methods.See for instance, Liu and He [15], Wu [23], Sun and Wu [21], Chen and Li [3], Li and Ye [8], He and Zou [6], Zhang et al. [26], Zhang and Zhang [27], Liu and Guo [12][13][14], Liu and Chen [4,11] and the references therein.
On the other hand, many papers concerned with the following quasilinear Schrödinger equation Such equations arise in various branches of mathematical physics and have been extensively studied in recent years.For example, the problem (1.4) was transformed to be a semilinear one by a change of variables and the existence of positive solutions of problem (1.4) in [18] was obtained on an Orlicz space by using the mountain pass theorem.The same method was also used in [5], but the usual Sobolev space H 1 (R N ) framework was used as the working space.Liu, Wang and Wang [19] obtained the existence of both one sign and nodal ground state type solutions of problem (1.4) by the Nehari method.In [17], the authors presented an approach to study problem (1.4) and proved that the solutions of problem (1.4) can be obtained as limits of 4-Laplacian perturbations.However, to the best of our knowledge, very few papers deal with problem (1.1) in the literature.More precisely, Liang and Shi [9] studied the problem (1.1) and obtained infinitely many solutions which tend to zero via a concentration-compactness principle and the minimax methods.In [24], the authors got the infinitely many small energy solutions of problem (1.1) by applying Clark's theorem.
Motivated by the reasons above, the aim of this paper is to show the existence of ground state solutions of problem (1.1).Different from the semilinear problems, the feature of the quasilinear problem is the appearance of the term [∆(u 2 )]u.This makes the problem more challenging and interesting because in general there is no suitable space in which the energy functional enjoys both smoothness and compactness.Therefore, the variational methods can not be applied directly.As we shall see in the present paper, problem (1.1) can be viewed as an elliptic equation coupled with a non-local term.The competing effect of the non-local term with the critical nonlinearity and the lack of compactness of the embedding of H 1 (R 3 ) into the space L p (R 3 ), prevent us from using the variational methods in a standard way.Following the idea of [5,18], we transform the problem to a semilinear one by a change of variables.Note that the problem (1.1) becomes problem (1.4) when b = 0.It is worth pointing out that although the idea was used to solve the problem (1.4) above, the adaptation to the procedure to our problem is not trivial at all since the appearance of non-local term.To obtain the ground state solution of problem (1.1), however, some more delicate estimates are needed in the present paper.
Before stating our main results we need to introduce some notations and definitions.
Remark 1.5.It should be mentioned that the authors in [16] have proved the problem (1.1) with b = 0 has no nontrivial solution if x • ∇V(x) ≥ 0 and p ≥ 11.This is the reason why we just consider the problem for 1 < p < 11.
Remark 1.6.Compared to the previous results (see e.g.[9]), the main novelty in this paper is that we are able to obtain the existence of the ground state solution of problem (1.1) with 1 < p < 11.Moreover, since we consider the critical case, our main results are also different from [24] in which the authors studied the nontrivial solutions.
The remainder of this paper is organized as follows.In Section 2, we present some preliminaries while the proofs of our main results is given in Section 3.

Preliminaries
Define the Hilbert space It is well known that the embedding X → L s (R 3 ) for 2 ≤ s < 6 is compact under the condition (V).Problem (1.1) is the Euler-Lagrange equation associated with the natural energy functional Unfortunately, I in general is not well defined on X because ∇(v 2 ) is not always in L 1 (R 3 ).To overcome this difficulty, based on the strategy developed in [5], we introduce a C ∞ -function f defined by Some properties of the function f are necessary in our arguments which we list below.The corresponding proofs can be found in [5,18,25].We omit them here.
Lemma 2.1.The function f enjoys the following properties: (1) f is uniquely defined and invertible C ∞ -function; (2) t is decreasing for t > 0; (6) There exists a positive constant C such that By making the change of variables v = f (u), the functional I can be rewritten as which is well defined on X.Moreover, by Lemma 2.1, standard arguments (see e.g.Proposition 1.12 in [22]) show that J ∈ C 1 (X, R) and for all u, ϕ ∈ X.As in [9], if u is a nontrivial critical point of J, then u is nontrivial solution of problem where and The following two lemmas show that the functional J has a mountain pass geometric structure.

Lemma 2.3.
There exists a u ∈ X such that J(u) < 0.

Proof of main results
Now, we are in the position to verify the main results.To this end, a further estimate of the mountain pass level value c is necessary.We recall that the best constant S for the Sobolev embedding Consider the function w ε defined by Let 0 < R < 1 and u ε = φw ε , where φ is a smooth cut-off function satisfying φ(x) = 1 for |x| ≤ R and φ(x) = 0 for |x| ≥ 2R.For any ε > 0, it is known that and the infimum in (3.1) is achieved by the function w 2 ε .Moreover, followed by [2], a direct computation yields that and Taking u ε as a test function, we have the following estimates for the level value c given in (2.6).(ii) If 1 < p ≤ 9, then there exists a constant µ * > 0 such that c < 1 6 S 3 2 for any µ > µ * .
Recall that, for any c ∈ R, we say {u n } is a (C) c sequence of J if J(u n ) → c and (1 + u n )J (u n ) → 0 as n → ∞.In order to obtain the existence of ground state solutions, we need to study some behaviors of a (C) c sequence of J carefully.Lemma 3.2.Let c ∈ R and {u n } ⊂ X be a (C) c sequence of J. Then {Φ(u n )} is bounded, where Φ is defined in (2.7).In particular, Hence, J (u n ), w n → 0 as n → ∞.
which means that In particular, by Lemma 2.1 (6) and (3.14), we have which together with (3.14) implies that {u n } is bounded in H 1 (R N ).The proof is completed.In what follows, we shall give the proof of Theorem 1.3 and Theorem 1.4.Since the proofs of them are similar, we just give the details of Theorem 1.3.
Proof of Theorem 1.3.Let c be the mountain pass level given in (2.6).From Lemma 2.2, Lemma 2.3 and the mountain pass theorem (see e.g.Theorem 3 in [20]), the functional J has a (C) c sequence {u n } ⊂ X.In view of Lemma 3.2, we may assume that u n u in H 1 (R3 ) and f (u n ) f (u) in X, which implies that u n → u in L s loc (R 3 ) for 2 < s < 6 and f (u n ) → f (u) in L s loc (R 3 ) for 2 < s < 12. Hence J (u n ), ϕ → J (u), ϕ = 0 for any ϕ ∈ C ∞ 0 (R 3 ), that is, u is a weak solution of (2.4).Moreover, since the embedding X → L 2 (R 3 ) is compact for 2 ≤ s < 6, we get f (u n ) → f (u) in L 2 (R 3 ) for 2 ≤ s < 6.We conclude from Lemma 3.1 (i) that c < 1 6 S

Notation 1.1. Throughout
this paper, we denote by • r the L r -norm, 1 ≤ r ≤ ∞, and we use the notation → ( ) to denote strong (weak) convergence.Also, if we take a subsequence of a sequence {u n } we shall denote it again {u n }.We use o(1) to denote any quantity which tends to zero when n → ∞.C and C i express distinct positive constants which may vary from line to line.