Solutions with prescribed mass for a critical Choquard equation driven by a local-nonlocal operator

Nidhi Nidhi
Konijeti Sreenadh

Abstract. In this paper, we study the normalized solutions of the following critical growth Choquard equation with mixed local and nonlocal operators: \[\begin{split}-\Delta u +(-\Delta)^s u &= \lambda u +\mu |u|^{p-2}u +(I_{\alpha}*|u|^{2^*_{\alpha}})|u|^{2^*_{\alpha}-2}u \quad\text{in}\quad \mathbb{R}^N,\\ \| u\|_2 &= \tau,\end{split}\] where \(N\geq 3\), \(\tau\gt 0\), \(I_{\alpha}\) 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 nonlocal fractional Laplacian operator with \(s\in (0,1)\), \(\mu\gt 0\) is a parameter and \(\lambda\) appears as Lagrange multiplier. We show the existence of at least two distinct solutions in the presence of the mass-subcritical perturbation \(\mu |u|^{p-2}u\) with \(2\gt p\gt 2+\frac{4s}{N}\) under some assumptions on \(\tau\).

Keywords: normalized solution, Choquard equation, critical exponent, mixed local and nonlocal operator, \(L^2\)-subcritical perturbation, nonlinear Schrödinger equation driven by local-nonlocal operator.

Mathematics Subject Classification: 35Q55, 35M10, 35J61, 35A01.

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