Continuous spectrum for a class of nonhomogeneous differential operators

TL;DR

This paper investigates the spectral properties of a class of nonhomogeneous differential operators with variable exponents, establishing the existence of a continuous spectrum of eigenvalues for certain parameter ranges.

Contribution

It introduces a new analysis of eigenvalues for a nonhomogeneous PDE with variable exponents, identifying a continuous spectrum and specific bounds for eigenvalues.

Findings

Existence of a continuous spectrum of eigenvalues for large λ.

Identification of bounds λ₀ and λ₁ for eigenvalues.

Non-existence of eigenvalues for λ below λ₀.

Abstract

We study the boundary value problem $-{\rm div}((|\nabla u|^{p_1(x)-2}+|\nabla u|^{p_2(x)-2})\nabla u)=\lambda|u|^{q(x)-2}u$ in $\Omega$, $u=0$ on $\partial\Omega$, where $\Omega$ is a bounded domain in $\RR^N$ with smooth boundary, $\lambda$ is a positive real number, and the continuous functions $p_1$, $p_2$, and $q$ satisfy $1<p_2(x)<q(x)<p_1(x)<N$ and $\max_{y\in\bar\Omega}q(y)<\frac{N p_2(x)}{N-p_2(x)}$ for any $x\in\bar\Omega$. The main result of this paper establishes the existence of two positive constants $\lambda_0$ and $\lambda_1$ with $\lambda_0\leq\lambda_1$ such that any $\lambda\in[\lambda_1,\infty)$ is an eigenvalue, while any $\lambda\in(0,\lambda_0)$ is not an eigenvalue of the above problem.