Minimax Theory and its Applications 04 (2019), No. 2, 217--229
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
[Abstract-pdf]
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^{s-1}-\mu u^{r-1}, \ \ \ &{\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 one-dimensional case via the time-mapping method.
Keywords: Sublinear elliptic problem, resonance, nonnegative solution, positive solution, minimax method, mountain pass, strong maximum principle.
MSC: 35J20, 35J25
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