Normalized Ground State Solutions for Nonautonomous Choquard Equations

Abstract

In this paper, we study normalized ground state solutions for the following nonautonomous Choquard equation

$$\left\{ {\matrix{{ - \Delta u - \lambda u = \left( {{1 \over {{{\left| x \right|}^\mu }}} * A{{\left| u \right|}^p}} \right)\,\,A{{\left| u \right|}^{p - 2}}u,} \hfill \cr {\int_{{\mathbb{R}^N}} {{{\left| u \right|}^2}{\rm{d}}x = c,\,\,\,\,\,u \in {H^1}({\mathbb{R}^N},\mathbb{R}),} } \hfill \cr } } \right.$$

where c > 0, 0 < μ < N, λ ∈ ℝ, A ∈ C¹ (ℝ^N, ℝ). For p ∈ (2_*,μ, ${\bar p}$, we prove that the Choquard equation possesses normalized ground state solutions, and the set of ground states is orbitally stable. For $p \in (\bar p,2_\mu ^ * )$, we find a normalized solution, which is not a global minimizer. 2*_μ and 2_*,μ are the upper and lower critical exponents due to the Hardy–Littlewood–Sobolev inequality, respectively. ${\bar p}$ is the L²-critical exponent. Our results generalize and extend some related results.