Electronic Journal of Qualitative Theory of Differential Equations

In this article, we study the following degenerated Schrödinger equations: { −∆γu + λV(x)u = f (x, u) in RN , u ∈ Eλ , where λ > 0 is a parameter, ∆γ is a degenerate elliptic operator, the potential V(x) has a potential well with bottom and the nonlinearity f (x, u) is either super-linear or sub-linear at infinity in u. The existence of ground state solution be obtained by using the variational methods.


Introduction
This article is concerned with a class of nonlinear Schrödinger equations: where λ > 0 is a parameter, ∆ γ is a degenerate elliptic operator of the form Corresponding author.Email: 11183356@qq.com Here, the functions γ j : R N → R are assumed to be continuous, different from zero and of class C 1 in R N \ Π, where Moreover, the function γ j satisfy the following properties: (i) There exists a semigroup of dilations {δ t } t>0 such that δ t : R N → R, δ t (x 1 , . . ., x N ) = (t ε 1 x 1 , . . ., t ε N x N ), γ j (δ t (x)) = t ε j −1 γ j (x), ∀x ∈ R N , ∀t > 0, j = 1, . . ., N.
The number is called the homogeneous dimension of R N with respect to {δ t } t>0 .
The ∆ γ -operator contains the following operator of Grušin-type where (x, y) denotes the point of R N 1 × R N 2 .This operator was studied by Grušin in [8] when α is an integer, and by Franchi and Lanconeli in [6,7], Loiudice in [11], Monti and Morbidelli in [13] when α is not an integer.The ∆ γ -operator also contains following semi-linear strongly degenerate operator where α, β are nonnegative real numbers.The P α,β -operator was studied in [1].For more information about the operator ∆ γ , please see [10].
In this paper, we study the existence of ground state solutions for the equation (1.1) under the assumptions that V is neither radially symmetric nor coercive.Precisely, we make the following assumptions.
(V2) There exists b > 0 such that the set {x ∈ R N : V(x) < b} is nonempty and has finite measure.
The conditions (V1) ∼ (V2) are special cases of steep potential well which were first introduced by Bartsch and Wang in [2].In recent years, steep potential well are widely used in various equation, such as Schrödinger equations, Schrödinger-Poisson equations and Klein-Gordon-Maxwell system and so on (see [2-4, 9, 14, 15]).Nextly, wee will require that the nonlinear term satisfies either the assumptions: or the assumptions: where 2 * γ := 2 Ñ Ñ−2 is the critical Sobolev exponent; |z| is an increasing function of z on R \ {0} for every x ∈ R N .Before stating our main results, we give several notations.For λ > 0, let Then, by assumption (V1), E λ is a Hilbert space with the inner product and norm respectively where Obviously, the embedding γ ] (see [12]).Thus for each 2 ≤ s ≤ 2 * γ , there exists d s > 0 such that where L s (R N ) denote a Lebesgue space, the norm in We point out that there are Rellich-type compact embeddings hold on bounded domains for subcritical exponents.By S 2 γ (Ω) we denote the set of all functions u ∈ L 2 (Ω) such that γ j ∂ x j u ∈ L 2 (Ω) for all j = 1, . . ., N, where Ω is a bounded domain with smooth boundary in R N .The space S 2 γ,0 (Ω) is defined as the closure of C 1 0 (Ω) in the space S 2 γ (Ω).We define the norm on this space as [10]).We can now state the main result: Theorem 1.1.Assume (V1) and ( f 1) ∼ ( f 2) are satisfied.Then ∀λ > 0, problem (1.1) admits at least a ground state solution in E λ .
Remark 1.2.To the best of our knowledge, it seems that Theorem 1.1 is the first result about the existence of ground state solutions for the semi-linear ∆ γ differential equation in R N .By the way, we would like to point out that in [12] the authors study existence of infinitely many solutions for semi-linear degenerate Schrödinger equations with the potential V(x) satisfying the coercivity condition which implies V2) and ( f 1) ∼ ( f 5) are satisfied.Then there exists Λ > 0 such that problem (1.1) has at least a ground state solution in E λ , for all λ > Λ.
Remark 1.4.We point out that the Schrödinger equation with general steep potential well is considered in reference [3,4], but they consider a special nonlinear term, where f (x, z) = |z| p−2 z(2 < p < 2 * ).At the same time, we also point out that although the Schrödinger equation with general steep potential well and the general nonlinear term are considered in reference [2,9], the nonlinear term there satisfies the following Ambrosetti-Rabinowitz type condition: (AR) There exist µ > 2 and L > 0, such that The nonlinear term we consider here is not required to satisfy the Ambrosetti-Rabinowitz type condition, for example we allow nonlinearities of the type By a simple calculation, we have and Now, it is easy to verify that the function f satisfies our assumptions and does not satisfy the Ambrosetti-Rabinowitz type condition.
To obtain our main results, we have to overcome some difficulties in our proof.The main difficulty consists in the lack of compactness of the Since we assume that the potential is not radially symmetric, we cannot use the usual way to recover compactness, for example, restricting in the subspace of radial functions of E λ .We also cannot borrow some ideas in [12] to recover compactness because the potential do not satisfied the coercivity condition.To recover the compactness, we establish the parameter dependent compactness conditions.Now, we define the following energy functional for any u ∈ E λ .It is well known that J λ is a C 1 functional with derivative given by for any u, v ∈ E λ .We have that u is a weak solution of equation 2 The proof of main results for f sub-linear at infinity in u Lemma 2.1 (see [17]).Let E be a real Banach space and J ∈ C 1 (E, R) satisfy the (PS) condition.If J is bounded from below, then c = inf E J is critical value of J.
Proof.It follows from ( f 1) that we can get The above inequality combined with the Hölder inequality and (1.2) shows that Lemma 2.3.Assume that (V1) and ( f 1) are satisfied, then J λ satisfies the (PS) condition for each λ > 0.
Proof.We suppose that {u n } is a Palais-Smale sequence of J λ , that is for some Therefore, up to a subsequence, there are u ∈ E λ , we have (2.3) By ( f 1) , for any fixed ε > 0, we can choose R ε > 0 such that It follows that (2.3), we obtain that Hence, there exists N 0 ∈ N such that we have Combing this with the Hölder inequality and ( f 1) , for any n ≥ N 0 we have that 1 2 • ε. (2.6) Again by ( f 1) , the Hölder inequality and (2.4), we obtain that (2.7) Since ε is arbitrary, by (2.6) and (2.7), we known that Thus, from (1.4) and ( 2.3), it holds Proof of Theorem 1.1.By Lemmas 2.1, 2.2 and 2.3, we known that Next, we will prove c λ = 0. Let u ∈ E λ and u λ = 1, by ( f 2) , we can get Since 1 < a 0 < 2, as t > 0 small enough, J λ (tu) < 0. Hence c λ = inf E λ J λ (u) < 0, equation (1.1) possesses at least a nontrivial ground state solution u λ for every λ > 0. Then the proof of Theorem 1.1 is completed.

The proof of main results for f super-linear at infinity in u
To complete the proof of our theorem, we need the following definition of Cerami condition and critical point theorem(see [16]).
If any sequence {u n } ⊂ H such that J(u n ) → c and J (u n )(1 + u n ) → 0, then this sequence is called a (C) c sequence.If any (C) c sequence {u n } ⊂ H of J has a convergent subsequence, then this C 1 functional J satisfies (C) c condition.

Theorem 3.1 (Mountain Pass Theorem).
Let H be a real Banach space and J ∈ C 1 (H, R).Assume that there exist v 0 ∈ H, v 1 ∈ H, and a bounded open neighborhood Ω of v 0 such that v 1 ∈ Ω and J(γ(t)).
If J satisfies the (C) c condition, then c is a critical value of J and c > max{J(v 0 ), J(v 1 )}.
It is clear that That is, the geometric conditions of mountain pass theorem are satisfied.Thus, the mountain pass value J λ (γ(t)). exists.
Proof.Let {u n } ⊂ E λ satisfies (3.5).By Lemma 3.4, we known that {u n } is bounded in E λ .Thus, up to a subsequence, we have that (3.17) (3.16), which implies that Next, by using the similar proof method of Proposition A.1 in the literature [5], we can get that and for any ϕ ∈ E λ .By (3.19) and (3.20), we can obtain that Combing with (3.21) and u n = v n + u, for any ϕ ∈ E λ we have that here Ω ϕ is the support set of ϕ.For each ϕ ∈ C ∞ 0 (R N ), by (3.16) we have (u n − u, ϕ) λ → 0, as n → ∞.
It is easy obtained that J λ (u) = max t≥0 J λ (tu) by ( f 5) for any u ∈ S. So, from the arbitrariness of u, we obtain inf u∈S J λ (u) ≥ c λ .
Thus, c λ = inf u∈S J λ (u), and we can conclude that u λ is the ground state solution, then the proof of Theorem 1.3 is completed.