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

Mihăilescu, Mihai

Czechoslovak Mathematical Journal, Tome 58 (2008) no. 1, pp. 155-172

Résumé

We study the boundary value problem $-{\mathrm div}((|\nabla u|^{p_1(x) -2}+|\nabla u|^{p_2(x)-2})\nabla u)=f(x,u)$ in $\Omega $, $u=0$ on $\partial \Omega $, where $\Omega $ is a smooth bounded domain in ${\mathbb{R}} ^N$. 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 \lbrace p_1(x),p_2(x)\rbrace $ for any $x\in \overline{\Omega }$ or $m(x)$ 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 ${\mathbb{Z}} _2$-symmetric version for even functionals of the Mountain Pass Theorem and some adequate variational methods.

Détail

Classification : 35D05, 35J60, 35J70, 47J30, 58E05, 68T40, 76A02, 76A05
Keywords: $p(x)$-Laplace operator; generalized Lebesgue-Sobolev space; critical point; weak solution; electrorheological fluid