Continuous spectrum for a class of nonhomogeneous differential operators

Abstract

We study the boundary value problem $-{\rm div}((|\nabla u|^{p_1(x)-2}+|\nabla u|^{p_2(x)-2})\nabla u)=λ|u|^{q(x)-2}u$ in $Ω$, $u=0$ on $\partialΩ$, where $Ω$ is a bounded domain in $\RR^N$ with smooth boundary, $λ$ 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 $\max_{y\in\barΩ}q(y)<\frac{N p_2(x)}{N-p_2(x)}$ for any $x\in\barΩ$. The main result of this paper establishes the existence of two positive constants $λ_0$ and $λ_1$ with $λ_0\leqλ_1$ such that any $λ\in[λ_1,\infty)$ is an eigenvalue, while any $λ\in(0,λ_0)$ is not an eigenvalue of the above problem.