Generalized spectrum of the $\boldsymbol{(p,2)}$-Laplacian under a parametric boundary condition

TL;DR

This paper investigates the eigenvalues of a $(p,2)$-Laplace operator with a nonlinear Steklov boundary condition, revealing a spectrum with an isolated zero eigenvalue and a continuous family of nonzero eigenvalues for all $p>1$.

Contribution

It extends the spectral analysis of the $(p,2)$-Laplace operator under nonlinear boundary conditions, identifying the structure of the eigenvalue set for all $p>1$.

Findings

Eigenvalue set includes an isolated zero eigenvalue.

Nonzero eigenvalues form a continuous family.

Results hold under certain integrability and boundedness conditions.

Abstract

In this paper we study an eigenvalue problem for the so called $(p,2)$-Laplace operator on a smooth bounded domain under a nonlinear Steklov type boundary condition, namely \begin{equation} \left\{ \begin{aligned} -\Delta_pu-\Delta u & =\lambda a(x)u \ \ \text{in}\ \Omega,\\ (|\nabla u|^{p-2}+1)\dfrac{\partial u}{\partial\nu} & =\lambda b(x)u \ \ \text{on}\ \partial\Omega . \end{aligned} \right. \end{equation} Under suitable integrability and boundedness assumptions on the positive weight functions $a$ and $b$, we show that, for all $p>1$, the eigenvalue set consists of an isolated null eigenvalue plus a continuous family of eigenvalues located away from zero.