Infinitely many non-radial solutions for a Choquard equation

Abstract

Abstract In this article, we consider the non-linear Choquard equation − Δ u + V ( ∣ x ∣ ) u = ∫ R 3 ∣ u ( y ) ∣ 2 ∣ x − y ∣ d y u in R 3 , -\Delta u+V\left(| x| )u=\left(\mathop{\int }\limits_{{{\mathbb{R}}}^{3}}\frac{| u(y){| }^{2}}{| x-y| }{\rm{d}}y\right)u\hspace{1.0em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{3}, where V ( r ) V\left(r) is a positive bounded function. Under some proper assumptions on V ( r ) V\left(r) , we are able to establish the existence of infinitely many non-radial solutions.