Existence and mass concentration of standing waves for inhomogeneous NLS equation with a bounded potential

This paper is concerned with the following minimization problem \(\begin{aligned} e_p(M)=\inf \{E_p(u):u \in H^1(\mathbb {R}^N),\Vert u\Vert ^2_{L^2}=M^2 \}, \end{aligned}\) $$\begin{array}{c}\hfill e_p(M)=inf\{E_p(u):u\in H^1({\mathbb{R}}^N){,\Vert u\Vert }_{L^2}^2=M^2\},\end{array}$$ where energy functional \(E_p(u)\) $E_p(u)$ is defined by \(\begin{aligned} E_p(u)=\Vert \nabla u \Vert _{L^2}^2 +\int _{\mathbb {R}^N} V(x)|u |^2dx -\frac{2}{p+2} \int _{\mathbb {R}^N}|x |^{-h} | u|^{p+2}dx \end{aligned}\) $$\begin{array}{c}\hfill E_p(u)={\Vert \mathrm{\nabla }u\Vert }_{L^2}^2+{\int }_{{\mathbb{R}}^N}{V(x)|u|}^{2}dx-\frac{2}{p+2}{\int }_{{\mathbb{R}}^N}{|x|}^{-h}{|u|}^{p+2}dx\end{array}$$ and V is a bounded potential. For \(0<p< p^*:=\frac{4-2\,h}{N}(0<h<\min \{2,N\})\) $0<p<p^{\ast }:=\frac{4-2h}{N}(0<h<min\{2,N\})$ , it is shown that there exists a constant \(M_0\ge 0\) $M_0\ge 0$ , such that the minimization problem exists at least one minimizer if \(M> M_0\) $M>M_0$ . When \(p=p^*,\) $p=p^{\ast },$ the minimization problem exists at least one minimizer if \(M\in (M_{*},\Vert Q_{p^*}\Vert _{L^2}),\) $M\in (M_{\ast },\Vert Q_{p^{\ast }}{\Vert }_{L^2}),$ where constant \(M_{*}\ge 0\) $M_{\ast }\ge 0$ and \(Q_{p^*}\) $Q_{p^{\ast }}$ is the unique positive radial solution of \(-\Delta u+u -| x|^{-h}|u |^{p^*} u=0,\) $-\mathrm{\Delta }u+u-{|x|}^{-h}{|u|}^{p^{\ast }}u=0,$ and under some assumptions on V, there is no minimizer if \(M\ge \Vert Q_{p^*}\Vert _{L^2}\) $M\ge \Vert Q_{p^{\ast }}{\Vert }_{L^2}$ . Moreover, when \(0<p<p^*,\) $0<p<p^{\ast },$ for fixed \(M> \Vert Q_{p^*}\Vert _{L^2}\) $M>\Vert Q_{p^{\ast }}{\Vert }_{L^2}$ , we analyze the concentration behavior of minimizers as \(p \nearrow p^* \) $p\nearrow p^{\ast }$ .