Existence and multiplicity of normalized solutions to a large class of elliptic equations on bounded domains with general boundary conditions

TL;DR

This paper investigates the existence and multiplicity of normalized solutions for a broad class of nonlinear Schrödinger equations on bounded domains with general boundary conditions, using a perturbation method.

Contribution

It introduces a novel perturbation approach to establish existence and multiplicity results for normalized solutions in various nonlinear Schrödinger equations with diverse boundary conditions.

Findings

Proves existence of solutions for the nonlinear Schrödinger equation with general boundary conditions.

Establishes multiplicity results under certain conditions.

Extends methods to equations with critical exponential growth, magnetic fields, biharmonic, and Choquard equations.

Abstract

In this paper, by adapting the perturbation method, we study the existence and multiplicity of normalized solutions for the following nonlinear Schr\"odinger equation $$ \left\{ \begin{array}{ll} -\Delta u = \lambda u + f(u)\quad & \text{in } \Omega, \mathcal{B}_{\alpha,\zeta,\gamma}u = 0 & \text{on } \partial \Omega, \int_{\Omega} |u|^2\,dx = \mu, \end{array} \right. \leqno{(P)^\mu_{\alpha,\zeta,\gamma}} $$ where $\Omega \subset \mathbb{R}^N$ ($N \geq 1$) is a smooth bounded domain, $\mu>0$ is prescribed, $\lambda \in \mathbb{R}$ is a part of the unknown which appears as a Lagrange multiplier, $f,g:\mathbb{R} \to \mathbb{R}$ are continuous functions satisfying some technical conditions. The boundary operator $\mathcal{B}_{\alpha,\zeta,\gamma}$ is defined by $$ \mathcal{B}_{\alpha,\zeta,\gamma}u=\alpha u+\zeta \frac{\partial u}{\partial \eta }-\gamma g(u), $$ where $\alpha,\zeta,\gamma \in \{0,1\}$ and $\eta$ denotes the outward unit normal on $\partial\Omega$. Moreover, we highlight several further applications of our approach, including the nonlinear Schr\"{o}dinger equations with critical exponential growth in $\mathbb{R}^{2}$, the nonlinear Schr\"{o}dinger equations with magnetic fields, the biharmonic equations, and the Choquard equations, among others.