Nontrivial solutions of some quasilinear problems via a cohomological local splitting

This paper deals with a generalization of the $p$-Laplacian type boundary value problem \[ \left\{\begin{array}{ll} - {\rm div} (p \bar A(x,u) |\nabla u|^{p-2}\nabla u) + \bar A_t(x,u) |\nabla u|^p = g(x,u) & \hbox{in $\Omega$,}\\ u = 0 & \hbox{on $\bdry{\Omega}$,} \end{array}\right. \] $\Omega$ being a bounded domain in $\R^N$. Under suitable assumptions and if $p > N$, the existence of a nontrivial solution can be proved by means of variational tools and a cohomological local splitting.