
Minimax Theory and its Applications 04 (2019), No. 2, 217229 Copyright Heldermann Verlag 2019 Existence Results and Strong Maximum Principle for a Resonant Sublinear Elliptic Problem Giovanni Anello Dept. of Mathematics and Computer Science, Physical Science and Earth Science, University of Messina, Viale F. Stagno d'Alcontres 31, 8166 Messina, Italy ganello@unime.it [Abstractpdf] Let $\Omega$ be a bounded smooth connected open set in $\mathbb{R}^N$ and let $\lambda_1$ be the first eigenvalue of the Laplacian on $\Omega$. We study the resonant elliptic problem \begin{eqnarray*} \left\{\begin{array}{lll} \Delta u=\lambda_1 u+u^{s1}\mu u^{r1}, \ \ \ &{\rm in}\ \ \ \Omega\\ u\geq 0, \ \ \ &{\rm in}\ \ \ \Omega\\ u_{\mid \partial \Omega}=0 \end{array}\right. \end{eqnarray*} where $s\in ]1,2[$, $r\in ]1,s[$, and $\mu\in ]0,+\infty[$. An existence result of nonzero solutions is established via minimax and perturbation methods. Furthermore, for $\mu$ large enough, we prove a Strong Maximum Principle for the solutions of this problem. In particular, we extend to higher dimension an analogous recent result obtained in the onedimensional case via the timemapping method. Keywords: Sublinear elliptic problem, resonance, nonnegative solution, positive solution, minimax method, mountain pass, strong maximum principle. MSC: 35J20, 35J25 [ Fulltextpdf (125 KB)] for subscribers only. 