Continuous spectrum for a class of nonhomogeneous differential operators

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$\Omega$isaboundeddomainin$\RR^N$withsmoothboundary,$\lambda$ 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 ${\max }_{y\in \bar{\mathrm{\Omega }}}q(y)<\frac{N{p}_2(x)}{N-p_2(x)}$ for any $x\in \bar{\mathrm{\Omega }}$. The main result of this paper establishes the existence of two positive constants ${\lambda }_0$ and ${\lambda }_1$ with ${\lambda }_0\le {\lambda }_1$ such that any $\lambda \in [{\lambda }_1,\mathrm{\infty })$ is an eigenvalue, while any $\lambda \in (0,{\lambda }_0)$ is not an eigenvalue of the above problem.