Standing waves with prescribed mass for NLS equations with Hardy potential in the half-space under Neumman boundary condition

Abstract

Consider the Neumann problem: \begin{eqnarray*} \begin{cases} &-\Delta u-\frac{\mu}{|x|^2}u +\lambda u =|u|^{q-2}u+|u|^{p-2}u ~~~\mbox{in}~~\mathbb{R}_+^N,~N\ge3, &\frac{\partial u}{\partial \nu}=0 ~~ \mbox{on}~~ \partial\mathbb{R}_+^N \end{cases} \end{eqnarray*} with the prescribed mass: \begin{equation*} \int_{\mathbb{R}_+^N}|u|^2 dx=a>0, \end{equation*} where $\mathbb{R}_+^N$ denotes the upper half-space in $\mathbb{R}^N$, $\frac{1}{|x|^2}$ is the Hardy potential, $2<q<2+\frac{4}{N}<p<2^*$, $\mu>0$, $\nu$ stands for the outward unit normal vector to $\partial \mathbb{R}_+^N$, and $\lambda$ appears as a Lagrange multiplier. Firstly, by applying Ekeland's variational principle, we establish the existence of normalized solutions that correspond to local minima of the associated energy functional. Furthermore, we find a second normalized solution of mountain pass type by employing a parameterized minimax principle that incorporates Morse index information. Our analysis relies on a Hardy inequality in $H^1(\mathbb{R}_+^N)$, as well as a Pohozaev identity involving the Hardy potential on $\mathbb{R}_+^N$. This work provides a variational framework for investigating the existence of normalized solutions to the Hardy type system within a half-space, and our approach is flexible, allowing it to be adapted to handle more general nonlinearities.