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} -Δu +(-Δ)^s u & = & λu +μ|u|^{p-2}u +(I_α*|u|^{2^*_α})|u|^{2^*_α-2}u \text{ in } \mathbb{R}^N;\;\; \left\| u \right\|_2 & = & τ, \end{array} \end{equation*} here $N\ge 3$, $\tau >0$, $I_{\alpha }$ is the Riesz potential of order $\alpha \in (0,N)$, $2_{\alpha }^{\ast }=\frac{N+\alpha }{N-2}$ 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 $\tau$.