Existence of Multiple Positive Solutions for Choquard Equation with Perturbation

Abstract EN

This paper is concerned with the following Choquard equation with perturbation: - Δ u + V ( x ) u = ( 1 / | x | α ∗ | u | p ) | u | p - 2 u + g ( x ) , u ∈ H 1 ( R N ) , where N ≥ 3 , α ∈ ( 0 , N ) , and 2 - ( α / N ) < p < ( 2 N - α ) / ( N - 2 ) .

This kind of equations is well known as the Choquard or nonlinear Schrödinger-Newton equation.

Under some assumptions for the functions V ( x ) , we prove the existence of multiple positive solutions of the equation.

Moreover, we also show that these results still hold for more generalized Choquard equation with perturbation.