Infinitely many solutions of degenerate quasilinear Schrödinger equation with general potentials

In this paper, we study the following quasilinear Schrödinger equation: −div(a(x,∇u))+V(x)|x|−αp∗|u|p−2u=K(x)|x|−αp∗f(x,u)in RN,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ -\operatorname{div}\bigl(a(x,\nabla u)\bigr)+V(x) \vert x \vert ^{-\alpha p^{*}} \vert u \vert ^{p-2}u=K(x) \vert x \vert ^{- \alpha p^{*}}f(x,u) \quad \text{in } \mathbb{R}^{N}, $$\end{document} where N≥3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$N\geq 3$\end{document}, 1<p<N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$1< p< N$\end{document}, −∞<α<N−pp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$-\infty <\alpha <\frac{N-p}{p}$\end{document}, α≤e≤α+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\alpha \leq e\leq \alpha +1$\end{document}, d=1+α−e\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$d=1+\alpha -e$\end{document}, p∗:=p∗(α,e)=NpN−dp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$p^{*}:=p^{*}(\alpha ,e)=\frac{Np}{N-dp}$\end{document} (critical Hardy–Sobolev exponent), V and K are nonnegative potentials, the function a satisfies suitable assumptions, and f is superlinear, which is weaker than the Ambrosetti–Rabinowitz-type condition. By using variational methods we obtain that the quasilinear Schrödinger equation has infinitely many nontrivial solutions.


Introduction
In this paper, we study the following quasilinear Schrödinger equation: (1.1) where N ≥ 3, 1 < p < N , -∞ < α < N-p p , α ≤ e ≤ α + 1, d = 1 + αe, p * := p * (α, e) = Np N-dp is the critical Hardy-Sobolev exponent, V and K are nonnegative potentials, f is of superlinear growth near infinity, and for some positive functions h 1 (x) ∈ L ∞ (R N ) and h 0 (x) ∈ L p/(p-1) α problem ⎧ ⎨ ⎩ -div(a(x, ∇u)) = f (x, u) in , For different types of a(x, ∇u), the quasilinear equation of the form (1.1) has been derived from several physical models. Especially, a(x, ∇u) = |∇u| p-2 ∇u and a(x, ∇u) = |x| -αp |∇u| p-2 ∇u were used for the problems of nonlinear diffusion, such as nonlinear optics, plasma physics, condensed matter physics, and so on. We refer the reader to [16,25] and references therein. This type of equation has been extensively studied in recent years with a huge variety of hypotheses on the potentials V (x) and K(x). For V bounded from below by a positive constant (V (x) ≥ V 1 > 0) and K(x) ≡ 1, we would like to cite [1,10,21] and references therein, and in case of K(x) ≡ 1, we refer to [18,23,25].
If V goes to zero as |x| → ∞, that is, which is called the zero mass case, we can cite [2,7,9], which use the same technique as that used in [2]. In the case where K vanishes at infinity, we refer to the papers in [3][4][5].
The cases of K bounded by a positive constant and unbounded K are considered in [14]. Finally, [11,12] deal with the comprehensive problems including the potentials V and K . In [3], with more general potentials K and V , the authors obtained an inequality of Hardy type and then the strong convergence in the whole space. As a matter of fact, they have obtained the compact embedding of E ⊂ D 1,2 (R N ) in L q K (R N ) with 2 < q < 2 * . Using the same way, the compact embedding of E ⊂ D 1,p In most of the aforesaid references, the Ambrosetti-Rabinowitz (AR) condition is usually assumed. It is very crucial to ensure the boundedness of the Palais-Smale (PS) sequences of the energy functional. However, there are many functions that do not satisfy the AR condition. So in this paper, to prove that there are infinitely many solutions to quasilinear Schrödinger equation, we develop a superquadratic condition, which is weaker than the condition AR.
There are many difficulties in solving the problem of relationship among nonlinearities, operator, and potentials. To overcome this, we prove the existence of infinitely many solutions to problem (1.1) with compact embedding by using Tang's methods in [24]. As far as we know, to prove the boundedness of the (C) c -sequence for problem (1.1), we must have compact embedding, so we need to enhance some conditions for potentials K(x) and V (x). Before proving our results, we need to make the following assumptions on a, A, V , K , and f .
(1) Functions a and A. We consider continuous functions a : ∂ξ . Let c 0 and c 1 be positive real numbers, and let h 0 (x) and h 1 (x) be nonnegative measurable real functions in R N such that h 0 (x) ∈ L p/(p-1) α (R N ) with α = αp p * and h 1 (x) ∈ L ∞ (R N ) with h 1 (x) ≥ 1 for a.e. x ∈ R N . We introduce the following hypotheses: The function A can be used in several cases. For example: with a suitable function θ . We get the operator div(|∇u| p-2 ∇u) + div(θ (x) ∇u √ 1+|∇u| 2 ), which can be regarded as the sum of the p-Laplacian operator and a degenerate-form mean-curvature operator.
Is easy to check thatlim |x|→∞ (3) Functions f and F. Let functions f : R N+1 → R and F : for all x ∈ R satisfy the following conditions: (f 1 ) there exist constants c 1 , c 2 > 0 and β ∈ (p, p * ) such that and there exist c 0 > 0 and κ > N dp such that (f 6 ) There exist μ > p and r 1 > 0 such that 19]) Is easy to check that the following nonlinearities f satisfy (f 1 ), (f 2 ), (f 4 ), and (f 6 ): Now we are ready to state the main theorems of this paper. It is easy to check that (f 1 ) and (f 6 ) imply (f 5 ). Thus we have the following corollary.
, and (f 6 ) seem to be weaker than the superquadratic conditions obtained in the aforementioned references.
Notations Considering α and K in equation (1.1), an open set B ⊂ R, and a measurable function u : B → R, we use the following notations.
• (1) terms that tend to zero as n → ∞. The weak ( ) and strong (→) convergences are always taken as n → ∞. • Hereafter C is a positive constant that can changes its value in a sequence of inequalities.
The remainder of the paper is organized as follows. In Sect. 2, we present variational framework. In Sect. 3, we state and prove the main results of the paper.

Variational framework
In this section, we want to use variational methods. So we define a convenient space and functional. We consider the spaces We define with k 0 given by the inequality A(x, ∇u) ≥ k 0 h 1 (x)(1 + |x| -αp )|∇u| p for all ξ ∈ R N and a.e.
x ∈ R N , which will be proved in Lemma 3.2. Evidently, E is continuously embedded into D 1,p α (R N ). By the weighted Caffarelli-Kohn-Nirenberg's inequality [13] R In E, we define the following energy functional J ∈ C 1 (E, R): Its Gateaux derivative is given by By condition (f 1 ) we have

Existence of infinitely many solutions
In this section, we prove the existence of infinitely many solutions for problem (1.1). Next, we give the definition of a (C) c -sequence. A sequence {u n } ⊂ X is said to be a (C) c -sequence if J(u n ) → c and J (u n ) (1 + u n ) → 0, and it is said to satisfy the (C) c -condition if any (C) c -sequence has a convergent subsequence.
To prove our results, we use the following symmetric mountain pass theorem.

1)
and there exists k 0 > 0 such that The following two lemmas discuss the continuous and compact embedding E → L q K,α (R N ) for all q ∈ [p, p * ).

Lemma 3.4 Let (VK1)-(VK2) be satisfied. Then E is compactly embedded in
V (x) → 0 as |x| → ∞. Hence for any ε > 0, there exists R > 0 such that K(x) ≤ εV (x) for |x| > R. Let {u n } ⊂ E be a bounded sequence of E. Going if necessary to a subsequence, we may assume that Next, we claim that Hence, for any ε > 0, we have from which (3.4) follows. Since |s| q /|s| p → 0 as s → 0 and |s| q /|s| p * → 0 as s → ∞, then for any ε > 0, there exists C > 0 such that To prove the lemma for general exponent q, we use an interpolation argument. Let u n → 0 in E. We have just proved that u n → 0 in L q K,α (R N ), that is, implying that u n → 0 in L q K,α (R N ). This completes the proof.
Next, we need the following lemmas to show that J satisfies Lemma 3.1.
is bounded in E.
Proof To prove the boundedness of {u n }, arguing by contradiction, suppose that u n → ∞ as n → ∞. Let v n = u n u n . Then v n = 1. Observe that for large n, It follows from (2.2) and (3.2) that (3.8) Passing to a subsequence, we may assume that v n v in E is satisfied. Then by Lemma 3.4, E is compactly embedded in L q K,α (R N ), q ∈ [p, p * ), v n → v in L q K,α (R N ), q ∈ [p, p * ), and v n → v a.e. on R N .
Lemma 3.7 Let p 1 , p 2 > 1, r ≥ 1, and ⊆ R N . Let g(x, t) be a Carathéodory function on × R satisfying (3.14). If u n → u in L K,α ( ) and u n → u for a.e. x ∈ , then Proof By Lemma 3.5 the sequence {u n } is bounded in E. Going if necessary to a subsequence, we can assume that u n u in E. By Lemma 3.4, u n → u in L q K,α (R N ) for q ∈ [p, p * ), which, together with Lemma 3.7, yields R N K(x)|x| -αp * f (x, u n )f (x, u) |u n -u| dx → 0, n → ∞. (3.20) Observe that J (u n ) -J (u), u nu = R N a x, ∇(u nu) ∇(u nu) + V (x)|x| -αp * |u n -u| p dx It is clear that J (u n ) -J (u), u nu → 0, n → ∞. A(x, ∇u n ) + 1 p V (x)|x| -αp * |u n | p dx -