Soliton solutions for fractional Schrödinger equations

Abstract

In this paper, we study the existence of a soliton type solution for the fractional Schrödinger equation ( − Δ ) s u + V ( x ) u + [ ( − Δ ) s u 2 ] u = λ | u | p − 1 u , u > 0 , x ∈ R N , where 0 < s < 1 , ( − Δ ) s denotes the fractional Laplacian of order s , N > 2 s , 2 s ∗ = 2 N N − 2 s , 1 < p < 2 s ∗ − 1 . We prove that the equation has a solution by using a constrained minimization argument.