Mountain pass type solutions for quasilinear elliptic equations

Abstract.

We establish the existence of weak solutions in an Orlicz-Sobolev space to the Dirichlet problem

$(D)\quad \left \{\begin{array}{rcll} -{\rm div} \left (a(|\nabla u(x)|)\nabla u(x)\right )& =& g(x,u), & \mbox{in} \Omega u& = &0, & \mbox{on} \partial\Omega, \end{array} \right .$where $\Omega$ is a bounded domain in ${\mathbb R}^N$, $g\in C(\overline{\Omega}\times\mathbb R,\mathbb R)$, and the function $\phi(s)= sa(|s|)$ is an increasing homeomorphism from ${\mathbb R}$ onto ${\mathbb R}$. Under appropriate conditions on $\phi$, $g$, and the Orlicz-Sobolev conjugate $\Phi_*$ of $\Phi(s)=\int_0^s\phi(t) dt,$ (conditions which reduce to subcriticality and superlinearity conditions in the case the functions are given by powers), we obtain the existence of nontrivial solutions which are of mountain pass type.