Normalized solutions to a critical growth Choquard equation involving mixed operators

Abstract

In this paper we study the existence and regularity results of normalized solutions to the following critical growth Choquard equation with mixed diffusion type operators: $$\begin{array}{r}\begin{array}{rl}& -\mathrm{\Delta }u+(-\mathrm{\Delta }{)}^{s}u=\lambda u+g(u)+(I_{\alpha }*|u{|}^{2_{\alpha }^{*}})|u{|}^{2_{\alpha }^{*}-2}u\mathrm{in}{\mathbb{R}}^N,\\ & {\int }_{{\mathbb{R}}^N}|u{|}^{2}dx={\tau }^2,\end{array}\end{array}$$ where $N⩾3$ , $\tau >0$ , $I_{\alpha }$ is the Riesz potential of order $\alpha \in (0,N)$ , $(-\mathrm{\Delta }{)}^s$ is the fractional laplacian operator, $2_{\alpha }^{*}=\frac{N+\alpha }{N-2}$ is the critical exponent with respect to the Hardy Littlewood Sobolev inequality, λ appears as a Lagrange multiplier and g is a real valued function satisfying some $L^2$ -supercritical conditions.