Existence of multiple normalized solutions to a critical growth Choquard equation involving mixed operator

Abstract

In this paper we study the normalized solutions of the following critical growth Choquard equation with mixed local and non-local operators: \begin{equation*} \begin{array}{rcl} -\Delta u +(-\Delta)^s u & = & \lambda u +\mu |u|^{p-2}u +(I_{\alpha}*|u|^{2^*_{\alpha}})|u|^{2^*_{\alpha}-2}u \text{ in } \mathbb{R}^N;\;\; \left\| u \right\|_2 & = & \tau, \end{array} \end{equation*} here $N\geq 3$, $\tau>0$, $I_{\alpha}$ is the Riesz potential of order $\alpha\in (0,N)$, $2^*_{\alpha}=\frac{N+\alpha}{N-2}$ is the critical exponent corresponding to the Hardy Littlewood Sobolev inequality, $(-\Delta)^s$ is the non-local fractional Laplacian operator with $s\in (0,1)$, $\mu>0$ is a parameter and $\lambda$ appears as Lagrange multiplier. We have shown the existence of atleast two distinct solutions in the presence of mass subcritical perturbation, $\mu |u|^{p-2}u$ with $2<p<2+\frac{4s}{N}$ under some assumptions on $\tau$.