Local uniqueness of minimizers for Choquard type equations

Abstract

We consider $L^2$-constraint minimizers of the Choquard energy functional with a trapping potential $V\left(x\right)={\left|x\right|}^2$. It is known that positive minimizers exist if and only if the parameter $a$ satisfies $a<a^{\ast }≔{\Vert Q\Vert }_2^2$, where $Q$ is the unique positive radial solution of $-\Delta u+u-{\left|u\right|}^{\frac{4}{3}}u=0$ in $R^3$. This paper focuses on the local uniqueness of minimizers by using energy estimates, blow-up analysis and establishing the Pohozăev identity.