Infinitely many non-radial solutions for a Choquard equation

Abstract

In this article, we consider the non-linear Choquard equation −Δu+V(∣x∣)u=∫R3∣u(y)∣2∣x−y∣dyuinR3,-\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.

Keywords

• choquard equation
• hardy-littlewood-sobolev inequality
• infinitely many non-radial solutions
• 35j20
• 35j60
• 35a15