Minimax Theory and its Applications 04 (2019), No. 1, 021--032
Copyright Heldermann Verlag 2019
A Multiplicity Result for a Non-Autonomous Sublinear Elliptic Problem Involving Nonlinearities Indefinite in Sign
Giovanni Anello
Dept. of Mathematical and Computer Science, University of Messina, Viale F. Stagno d'Alcontres 31, 98166 Messina, Italy
ganello@unime.it
Luca Furnari
Dept. of Mathematical and Computer Science, University of Messina, Viale F. Stagno d'Alcontres 31, 98166 Messina, Italy
lfurnari@unime.it
[Abstract-pdf]
\newcommand{\R}{{\mathbb R}} Let $\Omega$ be a bounded smooth domain in $\R^N$, let $\alpha,\beta \colon \Omega \rightarrow \R$ be two measurable functions, and let $s\in ]1,2[$ and $r\in ]1,s[$. We deal with the following non autonomous elliptic problem \begin{equation*} \left\{ \begin{aligned} & - \Delta u=\alpha(x)u^{s-1}-\mu \beta(x) u^{r-1}, \ \ \ &&{\rm in}\ \ \Omega\\ & u\geq 0, &&{\rm in}\ \ \Omega\\ & u_{\mid \partial \Omega}=0 \end{aligned} \right. \end{equation*} where $\mu\in \R$ is a parameter. We establish, via minimax methods, a multiplicity result under suitable summability conditions on the weight functions $\alpha,\beta$.
Keywords: Sublinear elliptic problem, weight function, nonnegative solution, positive solution, minimax method, mountain pass, multiplicity.
MSC: 35J20, 35J25.
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