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

Yuxuan Zhang, Xiaojun Chang, Lin Chen

TL;DR

This work addresses normalized (mass-constrained) standing waves for a nonlinear Schrödinger equation with a Hardy potential in the half-space under Neumann boundary conditions. It develops a robust variational framework on $\mathbb{R}_+^N$ by proving a Hardy inequality and a Pohozaev identity, then obtains two normalized solutions: a local minimizer via Ekeland's principle and a mountain-pass type solution through a Morse-index informed minimax with approximating problems $J_\rho$. A key contribution is the parameterized minimax approach that yields bounded PS sequences and clarifies the role of the Lagrange multiplier, with a careful compactness analysis that works for small $\mu$. The results provide a flexible method for normalized solutions to Hardy-type systems in half-spaces and can adapt to more general nonlinearities, advancing the understanding of constrained NLS problems with singular potentials.

Abstract

Consider the Neumann problem: \begin{eqnarray*} \begin{cases} &-Δu-\fracμ{|x|^2}u +λu =|u|^{q-2}u+|u|^{p-2}u ~~~\mbox{in}~~\mathbb{R}_+^N,~N\ge3, &\frac{\partial u}{\partial ν}=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, $2<q<2+\frac{4}{N}<p<2^*$, $μ>0$, $ν$ stands for the outward unit normal vector to $\partial \mathbb{R}_+^N$, and $λ$ 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.

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

TL;DR

This work addresses normalized (mass-constrained) standing waves for a nonlinear Schrödinger equation with a Hardy potential in the half-space under Neumann boundary conditions. It develops a robust variational framework on by proving a Hardy inequality and a Pohozaev identity, then obtains two normalized solutions: a local minimizer via Ekeland's principle and a mountain-pass type solution through a Morse-index informed minimax with approximating problems . A key contribution is the parameterized minimax approach that yields bounded PS sequences and clarifies the role of the Lagrange multiplier, with a careful compactness analysis that works for small . The results provide a flexible method for normalized solutions to Hardy-type systems in half-spaces and can adapt to more general nonlinearities, advancing the understanding of constrained NLS problems with singular potentials.

Abstract

Consider the Neumann problem: \begin{eqnarray*} \begin{cases} &-Δu-\fracμ{|x|^2}u +λu =|u|^{q-2}u+|u|^{p-2}u ~~~\mbox{in}~~\mathbb{R}_+^N,~N\ge3, &\frac{\partial u}{\partial ν}=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 denotes the upper half-space in , is the Hardy potential, , , stands for the outward unit normal vector to , and 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 , as well as a Pohozaev identity involving the Hardy potential on . 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.

Paper Structure

This paper contains 6 sections, 10 theorems, 104 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Existence of local minimizers
  4. Existence of mountain pass type solutions
  5. Approximating problems
  6. Original problem

Key Result

Theorem 1.1

Let $N\geq 3$, $q\in (2, 2_*)$, $p\in (2, 2^*)$. There exists a positive constant $a_*>0$ such that for any $a\in (0, a_*)$, there exists $\mu_a\in (0, \frac{\bar{\mu}}{4})$ such that when $\mu\in (0, \mu_a)$, problem start admits two normalized solutions, one of which is a local minimizer of $J|_{\

Theorems & Definitions (21)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 3.1
  • proof
  • Lemma 4.1
  • ...and 11 more