
Minimax Theory and its Applications 02 (2017), No. 1, 001007 Copyright 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 [Abstractpdf] 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 $$ \Delta_p u= (\lambda u^{s2}u u^{r2}u)\chi_{\{u\neq 0\}} $$ 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 [ Fulltextpdf (107 KB)] for subscribers only. 