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

This paper is concerned with the following minimization problem

$$\begin{aligned} e_p(M)=\inf \{E_p(u):u \in H^1(\mathbb {R}^N),\Vert u\Vert ^2_{L^2}=M^2 \}, \end{aligned}$$

where energy functional \(E_p(u)\) is defined by

$$\begin{aligned} E_p(u)=\Vert \nabla u \Vert _{L^2}^2 +\int _{\mathbb {R}^N} V(x)|u |^2dx -\frac{2}{p+2} \int _{\mathbb {R}^N}|x |^{-h} | u|^{p+2}dx \end{aligned}$$

and V is a bounded potential. For \(0<p< p^*:=\frac{4-2\,h}{N}(0<h<\min \{2,N\})\), it is shown that there exists a constant \(M_0\ge 0\), such that the minimization problem exists at least one minimizer if \(M> M_0\). When \(p=p^*,\) the minimization problem exists at least one minimizer if \(M\in (M_{*},\Vert Q_{p^*}\Vert _{L^2}),\) where constant \(M_{*}\ge 0\) and \(Q_{p^*}\) is the unique positive radial solution of \(-\Delta u+u -| x|^{-h}|u |^{p^*} u=0,\) and under some assumptions on V, there is no minimizer if \(M\ge \Vert Q_{p^*}\Vert _{L^2}\). Moreover, when \(0<p<p^*,\) for fixed \(M> \Vert Q_{p^*}\Vert _{L^2}\), we analyze the concentration behavior of minimizers as \(p \nearrow p^* \).