The effect of topology on the number of positive solutions of elliptic equation involving Hardy-Littlewood-Sobolev critical exponent

Abstract

In this article we are concern for the following Choquard equation \[ -Δu = λ|u|^{q-2}u +\left(\int_Ω\frac{|u(y)|^{2^*_μ}}{|x-y|^μ} dy \right)|u|^{2^*_μ-2} u \; \text{in}\; Ω,\quad u = 0 \; \text{ on } \partial Ω, \] where $\Omega$ is an open bounded set with continuous boundary in ${\mathbb{R}}^N(N\ge 3)$, $2_{\mu }^{\ast }=\frac{2N-\mu }{N-2}$ and $q\in [2,2^{\ast })$ where $2^{\ast }=\frac{2N}{N-2}$. Using Lusternik-Schnirelman theory, we associate the number of positive solutions of the above problem with the topology of $\Omega$. Indeed, we prove if $\lambda <{\lambda }_1$ then problem has ${\text{cat}}_{\Omega }(\Omega )$ positive solutions whenever $q\in [2,2^{\ast })$ and $N>3$ or $4<q<6$ and $N=3$.

21 pages