Normalized solutions to HLS upper critical focusing Choquard equation with a non-autonomous nonlocal perturbation

Abstract

This paper is concerned with the following HLS upper critical focusing Choquard equation with a non-autonomous nonlocal perturbation

$$\begin{aligned} {\left\{ \begin{array}{ll} -{\Delta }u-\mu (I_\alpha *[h|u|^p])h|u|^{p-2}u-(I_\alpha *|u|^{2^*_\alpha })|u|^{2^*_\alpha -2}u=\lambda u\ \ \text{ in }\ \mathbb {R}^N, \\ \int _{\mathbb {R}^N} u^2 dx = c, \end{array}\right. } \end{aligned}$$

where \(\mu ,c>0\), \(N \ge 3\), \(0<\alpha <N\), \(2_\alpha :=\frac{N+\alpha }{N}<p<2^*_\alpha :=\frac{N+\alpha }{N-2}\), \(\lambda \in \mathbb {R}\) is a Lagrange multiplier, \(I_\alpha \) is the Riesz potential and \(h:\mathbb {R}^N\rightarrow (0,\infty )\) is a continuous function. Under a class of reasonable assumptions on h, we prove the existence of normalized solutions to the above problem for the case \(\frac{N+\alpha +2}{N}\le p<\frac{N+\alpha }{N-2}\) and discuss its asymptotical behaviors as \(\mu \rightarrow 0^+\) and \(c\rightarrow 0^+\) respectively. When \(\frac{N+\alpha }{N}<p<\frac{N+\alpha +2}{N}\), we obtain the existence of one local minimizer after considering a suitable minimization problem.