Normalized solutions to Choquard equation including the critical exponents and a logarithmic perturbation

Abstract

We study the existence of normalized solutions to the following nonlinear Choquard equation(0.1)$-\Delta u+\lambda u=u\log u^2+\mu (I_{\alpha }⁎|u{|}^p\left)\right|u{|}^{p-2}u\text{in}{R}^N$ under the mass constraint(0.2)$\underset{R^N}{\int }{u}^{2}dx=c^2,$ where $c>0$ is a constant, $N\ge 3$, $0<\alpha <N$, $\mu >0$, $\frac{N+\alpha }{N}\le p\le \frac{N+\alpha }{N-2}$, and the parameter $\lambda \in R$ appears as a Lagrange multiplier. Under different assumptions on p and c, we first show the existence of the associated global minimizer which must be a ground state solution of (0.1) with the mass constraint (0.2) if $\frac{N+\alpha }{N}\le p\le \frac{N+\alpha +2}{N}$, and then we prove the existence of ground state solution and mountain-pass solution for (0.1) with the mass constraint (0.2) if $\frac{N+\alpha +2}{N}<p\le \frac{N+\alpha }{N-2}$, where $\frac{N+\alpha }{N-2}$ is the Hardy-Littlewood-Sobolev upper critical exponent, $\frac{N+\alpha }{N}$ is the Hardy-Littlewood-Sobolev lower critical exponent and $\frac{N+\alpha +2}{N}$ is the $L^2$-mass constraint critical exponent. Moreover, the asymptotic behaviors of ground state solution and mountain-pass solution as $c\to 0^{+}$ are obtained.