Limit profiles and the existence of bound-states in exterior domains for fractional Choquard equations with critical exponent

Abstract

This article is devoted to studying the existence of positive solutions to the following fractional Choquard equation: (−Δ)su+u=∫Ω∣u(y)∣p∣x−y∣N−αdy∣u∣p−2u+ε∫Ω∣u(y)∣2α,s*∣x−y∣N−αdy∣u∣2α,s*−2u,inΩ,u=0,onRN\Ω,\left\{\begin{array}{ll}{\left(-\Delta )}^{s}u+u=\left(\mathop{\displaystyle \int }\limits_{\Omega }\frac{{| u(y)| }^{p}}{{| x-y| }^{N-\alpha }}{\rm{d}}y\right){| u| }^{p-2}u+\varepsilon \left(\mathop{\displaystyle \int }\limits_{\Omega }\frac{{| u(y)| }^{{2}_{\alpha ,s}^{* }}}{{| x-y| }^{N-\alpha }}{\rm{d}}y\right){| u| }^{{2}_{\alpha ,s}^{* }-2}u,& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,\\ u=0,& \hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N}\backslash \Omega \right,\end{array}\right. where Ω\Omega is an exterior domain with smooth boundary ∂Ω≠∅\partial \Omega \ne \varnothing such that RN\Ω{{\mathbb{R}}}^{N}\backslash \Omega is bounded, N>2s,20\varepsilon \gt 0 is a parameter. We establish the limit profiles and uniqueness of positive radial ground-states for the limit equation without the critical exponent as α\alpha sufficiently close to NN. Then, combining variational method, barycentric functions, and Brouwer degree theory, we determine the existence of positive bound-state solutions provided that ε>0\varepsilon \gt 0 is sufficiently small.

Keywords

• fractional choquard equation
• uniqueness
• exterior domain
• critical exponent
• 35j65
• 35q55
• 35b33