Infinitely many solutions of degenerate quasilinear Schrödinger equation with general potentials

Abstract

Abstract In this paper, we study the following quasilinear Schrödinger equation: − div ( a ( x , ∇ u ) ) + V ( x ) | x | − α p ∗ | u | p − 2 u = K ( x ) | x | − α p ∗ f ( x , u ) in R N , $$ -\operatorname{div}\bigl(a(x,\nabla u)\bigr)+V(x) \vert x \vert ^{-\alpha p^{*}} \vert u \vert ^{p-2}u=K(x) \vert x \vert ^{- \alpha p^{*}}f(x,u) \quad \text{in } \mathbb{R}^{N}, $$ where N ≥ 3 $N\geq 3$ , 1 < p < N $1< p< N$ , − ∞ < α < N − p p $-\infty <\alpha <\frac{N-p}{p}$ , α ≤ e ≤ α + 1 $\alpha \leq e\leq \alpha +1$ , d = 1 + α − e $d=1+\alpha -e$ , p ∗ : = p ∗ ( α , e ) = N p N − d p $p^{*}:=p^{*}(\alpha ,e)=\frac{Np}{N-dp}$ (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.

Keywords

• Schrödinger equation
• Superlinear condition
• Infinitely many solutions