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

Abstract

Consider the Neumann problem: $\left(\begin{array}{cc}& -\Delta u-\frac{\mu }{{\left|x\right|}^2}u+\lambda u={\left|u\right|}^{q-2}u+{\left|u\right|}^{p-2}u\text{in}{R}_{+}^N,N\ge 3,\\ & \frac{\partial u}{\partial \nu }=0\text{on}\partial R_{+}^N\end{array}\right)$with the prescribed mass: ${\int }_{R_{+}^N}{\left|u\right|}^{2}dx=a>0,$where $R_{+}^N$ denotes the upper half-space in $R^N$, $\frac{1}{{\left|x\right|}^2}$ is the Hardy potential, $2<q<2+\frac{4}{N}<p<2^{\ast }$, $\mu >0$, $\nu$ stands for the outward unit normal vector to $\partial R_{+}^N$, 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 $H^1\left(R_{+}^N\right)$, as well as a Pohozaev identity involving the Hardy potential on $R_{+}^N$. 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.