Normalized solution to the Sch\"odinger equation with potential and general nonlinear term: Mass super-critical case

In present paper, we prove the existence of solutions $(\lambda, u)\in \R\times H^1(\R^N)$ to the following Schr\"odinger equation $$\begin{cases} -\Delta u(x)+V(x)u(x)+\lambda u(x)=g(u(x))\quad &\hbox{in}~\R^N\\ 0\leq u(x)\in H^1(\R^N), N\geq 3 \end{cases}$$ satisfying the normalization constraint $\displaystyle \int_{\R^N}u^2 dx=a$. 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 $$\inf\left\{\int \left[\frac{1}{2}|\nabla u|^2+\frac{1}{2}V(x)u^2-G(u)\right]dx : \|u\|_{L^2(\R^N)}^{2}=a, P[u]=0\right\},$$ where $G(s)=\int_0^s g(\tau)d\tau$ and $$P[u]=\int\left[|\nabla u|^2-\frac{1}{2}\langle \nabla V(x), x\rangle u^2 -N\left(\frac{1}{2}g(u)u-G(u)\right)\right]dx.$$