Czechoslovak Mathematical Journal, Vol. 58, No. 1, pp. 155-172, 2008

On a class of nonlinear problems involving a $p(x)$-Laplace type operator

Mihai Mihailescu

Mihai Mihailescu, Department of Mathematics, University of Craiova, 200 585 Craiova, Romania, e-mail:; Department of Mathematics, Central European University, 1051 Budapest, Hungary

Abstract: We study the boundary value problem $- div((|\nabla u|^{p_1(x) -2}+|\nabla u|^{p_2(x)-2})\nabla u)=f(x,u)$ in $\Omega$, $u=0$ on $\partial\Omega$, where $\Omega$ is a smooth bounded domain in $\RR^N$. Our attention is focused on two cases when $f(x,u)=\pm(-\lambda|u|^{m(x)-2}u+|u|^{q(x)-2}u)$, where $m(x)=\max\{ p_1(x),p_2(x)\}$ for any $x\in\overline\Omega$ or $m(x)<q(x)< \frc{N\cdot m(x)}{(N-m(x))}$ for any $x\in\overline\Omega$. In the former case we show the existence of infinitely many weak solutions for any $\lambda>0$. In the latter 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 Theorem and some adequate variational methods.

Keywords: $p(x)$-Laplace operator, generalized Lebesgue-Sobolev space, critical point, weak solution, electrorheological fluid

Classification (MSC 2000): 35D05, 35J60, 35J70, 58E05, 68T40, 76A02

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