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

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.