Variational versus pseudomonotone operator approach in parameter-dependent nonlinear elliptic problems

We study the existence of nontrivial solutions of para-meter-dependent quasilinear elliptic Dirichlet problems of the form%$$-Delta_pu=lambda f(u)quadmbox{in } Omega,quad u=0quadmbox{on } partialOmega,$$in a bounded domain $Omegasubset mathbb{R}^N$ with sufficiently smooth boundary, where $lambda$ is a real parameter and $Delta_p$ denotes the $p$-Laplacian. Recently the authors obtained multiplicity results by employing an abstract localization principle of critical points of functionals of the form $mathbb{E}=Phi-lambdaPsi$ on open sublevels of $Phi$, i.e., of sets of the form $Phi^{-1}(-infty, r)$, combined with differential inequality techniques and topological arguments. Unlike in those recent papers by the authors, the approach in this paper is based on pseudomonotone operator theory and fixed point techniques. The obtained results are compared with those obtained via the abstract variational principle. Moreover, by applying truncation techniques and regularity results we are able to deal with elliptic problems that involve discontinuous nonlinearities without making use of nonsmooth ana-lysis methods.