Normalized solutions for fractional Choquard equation with critical growth on bounded domain

Abstract

In this work, we establish the multiplicity of positive solutions for the following critical fractional Choquard equation with a perturbation on the star-shaped bounded domain $$ \left\{ \begin{array}{lr} (-\Delta)^s u = \lambda u +\alpha|u|^{p-2}u+ \left( \int\limits_{\Omega} \frac{|u(y)|^{2^{*}_{\mu ,s}}}{|x-y|^ \mu}\, dy\right) |u|^{2^{*}_{\mu ,s}-2}u\; \text{in} \; \Omega,\\ u>0\; \text{in}\; \Omega,\; \\ u = 0\; \text{in} \; \mathbb{R}^{N}\backslash\Omega, \\ \int_{\Omega}|u|^2 dx=d, \end{array} \right. $$ where, $s\in(0,1), N>2s$, $\alpha\in \mathbb{R}$, $d>0$, $2<p<2^*_s:=\frac{2N}{N-2s}$ and $2^{*}_{\mu ,s}:=\frac{2N-\mu}{N-2s}$ represents fractional Hardy-Littlewood-Sobolev critical exponent. Using the minimization technique over an appropriate set and the uniform mountain pass theorem, we prove the existence of first and second solutions, respectively.