Choquard equations with recurrent potentials

Abstract

In this article, we are concerned with the existence of nontrivial solutions to the Choquard equation −Δu+α(x)u=(∣x∣−μ∗∣u∣q)∣u∣q−2uinRN(N≥2),-\Delta u+\alpha \left(x)u=\left({| x| }^{-\mu }\ast {| u| }^{q}){| u| }^{q-2}u\hspace{1.0em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N}\hspace{1em}\left(N\ge 2), with recurrent potential α\alpha , where 0<μ<N0\lt \mu \lt N and 2N−μN<q<2N−μN−2\frac{2N-\mu }{N}\lt q\lt \frac{2N-\mu }{N-2}. Our results include some classical cases where α\alpha is constant and α\alpha is periodic, as well as some new cases, such as α\alpha being almost periodic and α\alpha being only bounded and uniformly continuous.

Keywords

• choquard equation
• recurrent potential
• almost periodic function
• 35j20
• 35j60
• 47g20