Journal Home Page Cumulative Index List of all Volumes Complete Contentsof this Volume Minimax Theory and its Applications 02 (2017), No. 1, 001--007Copyright Heldermann Verlag 2017 Elliptic Problems with non Lipschitz Nonlinearities: Some Recent Results and Open Questions Giovanni Anello Dept. of Mathematics, University of Messina, Viale F. Stagno d'Alcontres 31, 98166 S. Agata, Messina, Italy ganello@unime.it [Abstract-pdf] Let $p\in ]1,+\infty[$, let $r,s\in ]0,p[$, with $r < s$, and let $\lambda\in ]0,+\infty[$. In this paper, we present some recent existence and multiplicity results on the solutions of the Dirichlet problem for the elliptic equation $$-\Delta_p u= (\lambda |u|^{s-2}u- |u|^{r-2}u)\chi_{\{u\neq 0\}}$$ in a bounded domain $\Omega \subset \mathbb{R}^N$, with $0$-boundary data. Some related open questions are also proposed. Keywords: Elliptic boundary value problem, nonnegative solution, positive solution, least energy solution, least energy nodal solution, variational methods, indefinite nonlinearities. MSC: 35J20, 35J25 [ Fulltext-pdf  (107  KB)] for subscribers only.