Existence and multiplicity of normalized solutions to a large class of elliptic equations on bounded domains with general boundary conditions (2507.04624v1)

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.