On the first eigenvalue of a nonlinear Schr\"odinger type equation

TL;DR

This paper investigates the first eigenvalue of a nonlinear Schr"odinger type operator with Robin boundary conditions, analyzing how it depends on potential, boundary parameters, and domain shape, including differentiability and monotonicity properties.

Contribution

It provides new results on the differentiability of the first eigenvalue with respect to boundary parameters and domain shape, along with explicit formulas for these derivatives.

Findings

Established properties of the smallest eigenvalue with respect to potential.

Proved differentiability of the eigenvalue concerning Robin boundary parameter and derived explicit formulas.

Analyzed domain monotonicity properties of the first eigenvalue.

Abstract

We consider an eigenvalue problem for the generalized nonlinear Schr\"{o}dinger type operator with the Robin boundary condition as given below. \begin{equation*} \label{ab-Robin p-Laplace evp with potential term_intro} \left\{ \begin{split} -\Delta_p u+V(x)|u|^{p-2}u&=\lambda |u|^{p-2}u\quad &&\mathrm{in} ~\Omega,\\ |\nabla u|^{p-2}\frac{\partial u}{\partial\eta}+\beta|u|^{p-2}u&=0\quad &&\mathrm{on}~\partial\Omega, \end{split} \right. \end{equation*} where $\Delta_p u := \operatorname{div}(|\nabla u|^{p-2}\nabla u)$ is the $p$-Laplace operator, $\Omega $ is a bounded domain in $\mathbb{R}^n$ with smooth boundary, $V \in C^1(\mathbb{R}^n),$ $ \eta $ denotes the outward unit normal, and $ \beta $ is a positive real constant. We study the properties of its first eigenvalue with respect to the potential $V$, the boundary parameter $\beta$ as well as the domain. First, we establish some properties of the smallest eigenvalue $\lambda_1(V)$ with respect to the potential. We then prove the differentiability of $\lambda_1(V)$ with respect to the Robin boundary parameter $\beta$ and give an explicit formula for this derivative, which is then used to investigate some monotonicity properties of $\lambda_1(V).$ We also obtain a shape derivative formula for the smallest eigenvalue. Using these derivatives, we also study domain monotonicity properties of the first eigenvalue.