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 $N\geq 3$, $τ>0$, $I_α$ is the Riesz potential of order $α\in (0,N)$, $2^*_α=\frac{N+α}{N-2}$ is the critical exponent corresponding to the Hardy Littlewood Sobolev inequality, $(-Δ)^s$ is the non-local fractional Laplacian operator with $s\in (0,1)$, $μ>0$ is a parameter and $λ$ appears as Lagrange multiplier. We have shown the existence of atleast two distinct solutions in the presence of mass subcritical perturbation, $μ|u|^{p-2}u$ with $2<p<2+\frac{4s}{N}$ under some assumptions on $τ$.

Theorems & Definitions (23)

• proposition 1
• Definition 1.1
• Theorem 1.1
• Theorem 1.2
• lemma 1
• proof
• lemma 2
• proof
• lemma 3
• proof