Normalized solution to the Schödinger equation with potential and general nonlinear term: Mass super-critical case

Abstract

In present paper, we prove the existence of solutions $(\lambda ,u)\in \mathbb{R}\times H^1({\mathbb{R}}^N)$ to the following Schrödinger equation $$\{\begin{array}{ll}-\Delta u(x)+V(x)u(x)+\lambda u(x)=g(u(x))& in{\mathbb{R}}^N\\ 0\le u(x)\in H^1({\mathbb{R}}^N),N\ge 3& \end{array}$$ satisfying the normalization constraint ${\int }_{{\mathbb{R}}^N}{u}^{2}dx=a$. We treat the so-called mass super-critical case here. Under an explicit smallness assumption on $V$ and some Ambrosetti-Rabinowitz type conditions on $g$, we can prove the existence of ground state normalized solutions for prescribed mass $a>0$. Furthermore, we emphasize that the mountain pass characterization of a minimizing solution of the problem $$\inf \left\{\int \left[\frac{1}{2}|\nabla u{|}^2+\frac{1}{2}V(x)u^2-G(u)\right]dx:\Vert u{\Vert }_{L^2({\mathbb{R}}^N)}^2=a,P[u]=0\right\},$$ where $G(s)={\int }_0^{s}g(\tau )d\tau$ and $$P[u]=\int \left[|\nabla u{|}^2-\frac{1}{2}⟨\nabla V(x),x⟩u^2-N\left(\frac{1}{2}g(u)u-G(u)\right)\right]dx.$$