Infinitely many solutions for a class of fractional Schrödinger equations with sign-changing weight functions

Abstract

Abstract In this paper, we study the fractional Schrödinger equation { ( − Δ ) s u + u = a ( x ) | u | p − 2 u + b ( x ) | u | q − 2 u , u ∈ H s ( R N ) , $$ \textstyle\begin{cases} (-\Delta )^{s}u+u=a(x) \vert u \vert ^{p-2}u+b(x) \vert u \vert ^{q-2}u, \\ u\in H^{s}(\mathbb{R}^{N}), \end{cases} $$ where ( − Δ ) s $(-\Delta )^{s}$ denotes the fractional Laplacian of order s ∈ ( 0 , 1 ) $s\in (0,1)$ , N > 2 s $N>2s$ , 2 < p < q < 2 s ∗ $2< p< q<2^{*}_{s}$ , and 2 s ∗ $2^{*}_{s}$ is the fractional critical Sobolev exponent. The weight potentials a or b is a sign-changing function and satisfies some valid assumptions. We obtain the existence of infinitely many solutions to the problem by the Nehari manifold.

Keywords

• Fractional Schrödinger equation
• Sign-changing weight functions
• Nehari manifold