Infinitely many bound states for Choquard equations with local nonlinearities. (December 2019) - Li, Xinfu; Liu, Xiaonan; Ma, Shiwang
Abstract: In this paper, we consider the following Choquard equation ( CH ) − Δ u + u = ( I α ∗ | u | p ) | u | p − 2 u + V ( x ) | u | q − 2 u in R N, u ∈ H 1 ( R N ), where N ≥ 3, α ∈ ( ( N − 4 ) +, N ), p ∈ [ 2, N + α N − 2 ), q ∈ ( 2, 2 N N − 2 ) ∩ ( 1 + ( p − 1 ) ( N − 2 ) α N 2, 1 + 2 N ( N − α ) + N 2 ( p − 1 ) ( N − 2 ) α ) and I α is the Riesz potential. Under some suitable decay assumptions but without any symmetry property on V ( x ), we prove that the problem has infinitely many solutions, whose energy can be arbitrarily large.