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

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.