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.
Categories: math.AP