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Minimax Theory and its Applications 02 (2017), No. 1, 001--007
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


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

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