Full description of the eigenvalue set of the $(p,q)$-Laplacian with a Steklov-like boundary condition

TL;DR

This paper fully characterizes the eigenvalue set of a $(p,q)$-Laplacian problem with Steklov-like boundary conditions in smooth bounded domains, extending previous partial results and covering new parameter ranges.

Contribution

It provides a complete description of the eigenvalues for the $(p,q)$-Laplacian with Steklov boundary conditions, including cases not previously analyzed.

Findings

Complete eigenvalue set description for the $(p,q)$-Laplacian

Extension to new parameter ranges including $q<2$

Complementary to prior partial results

Abstract

In this paper we consider in a bounded domain $\Omega \subset \mathbb{R}^N$ with smooth boundary an eigenvalue problem for the negative $(p,q)$-Laplacian with a Steklov-like boundary condition, where $p,\, q\in (1,\infty)$, $p\neq q$, including the open case $p\in (1,\infty)$, $q\in (1, 2)$, $p\neq q$. A full description of the set of eigenvalues of this problem is provided. Our results complement those previously obtained by Abreu and Madeira \cite{AM}, Barbu and Moro\c{s}anu \cite{BM}, F\u{a}rc\u{a}\c{s}eanu, Mih\u{a}ilescu and Stancu-Dumitru \cite{FMS}, Mih\u{a}ilescu \cite{MMih}, Mih\u{a}ilescu and Moro\c{s}anu \cite{MM}.