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, $N\ge 3$, $2\le p<N$, $\tau >0$, $I_{\alpha }$ is the Riesz potential of order $\alpha \in (\max \{0,N-2p\},N)$, $p_{\alpha }^{\ast }=\frac{p}{2}\left(\frac{N+\alpha }{N-p}\right)$ is the critical exponent corresponding to the Hardy Littlewood Sobolev inequality, $(-{\Delta }_p{)}^s$ is the non-local fractional p-Laplacian operator with $s\in (0,1)$, $\mu >0$ is a parameter and $\lambda$ appears as a Lagrange multiplier. We show the existence of at least two distinct solutions in the presence of a mass subcritical perturbation, $\mu |u{|}^{q-2}u$ with $p<q<p+\frac{s{p}^2}{N}$ under some conditions on $p,N$ and $s$.