Three solutions for a fractional Schrödinger equation with vanishing potentials

Abstract

In this paper, we study the following fractional Schrödinger equation ( − Δ ) s u + V ( x ) u = K ( x ) f ( u ) + λ W ( x ) | u | p − 2 u , x ∈ R N , where λ > 0 is a parameter, ( − Δ ) s denotes the fractional Laplacian of order s ∈ ( 0 , 1 ) , N > 2 s , W ∈ L 2 2 − p ( R N , R + ) , 1 < p < 2 , V , K are nonnegative continuous functions and f is a continuous function with a quasicritical growth. Under some mild assumptions, we prove that the above equation has three solutions.