Multiplicity of solutions with prescribed mass for a quasilinear critical Choquard equation driven by a local-nonlocal 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} -Δ_p u +(-Δ_p)^s u & = & λ|u|^{p-2}u +μ|u|^{q-2}u +(I_α*|u|^{p^*_α})|u|^{p^*_α-2}u \text{ in } \mathbb{R}^N; \left\| u \right\|_p & = & τ. \end{array} \end{equation*} Here, $N\geq 3$, $2 \le p<N$, $τ>0$, $I_α$ is the Riesz potential of order $α\in (\max\{0,N-2p\}, N)$, $p^*_α=\frac{p}{2}\left(\frac{N+α}{N-p}\right)$ is the critical exponent corresponding to the Hardy Littlewood Sobolev inequality, $(-Δ_p)^s$ is the non-local fractional p-Laplacian operator with $s\in (0,1)$, $μ>0$ is a parameter and $λ$ appears as a Lagrange multiplier. We show the existence of at least two distinct solutions in the presence of a mass subcritical perturbation, $μ|u|^{q-2}u$ with $p<q<p+\frac{sp^2}{N}$ under some conditions on $p,N$ and $s$.