Nonlinear eigenvalue problems in Sobolev spaces with variable exponent

Abstract

We study the boundary value problem $-{\rm div}((|\nabla u|^{p\_1(x) -2}+|\nabla u|^{p\_2(x)-2})\nabla u)=f(x,u)$ in $\Omega$, $u=0$ on $\partial\Omega$, where $\Omega$ is a smooth bounded domain in $\RR^N$. We focus on the cases when $f\_\pm (x,u)=\pm(-\lambda|u|^{m(x)-2}u+|u|^{q(x)-2}u)$, where $m(x):=\max\{p\_1(x),p\_2(x)\} < q(x) < \frac{N\cdot m(x)}{N-m(x)}$ for any $x\in\bar\Omega$. In the first case we show the existence of infinitely many weak solutions for any $\lambda>0$. In the second case we prove that if $\lambda$ is large enough then there exists a nontrivial weak solution. Our approach relies on the variable exponent theory of generalized Lebesgue-Sobolev spaces, combined with a $\ZZ\_2$-symmetric version for even functionals of the Mountain Pass Lemma and some adequate variational methods.