## Topological Methods in Nonlinear Analysis

### A variable exponent diffusion problem of concave-convex nature

#### Abstract

We deal with the problem $$\begin{cases} -\Delta u = \lambda u^{q(x)} & \text{if } x\in \Omega,\\ u = 0 & \text{if } x\in \p\Omega, \end{cases}$$ where $\Omega\subset \mathbb R^N$ is a bounded smooth domain, $\lambda> 0$ is a parameter and the exponent $q(x)$ is a continuous positive function that takes values both greater than and less than one in $\overline{\Omega}$. It is therefore a kind of concave-convex problem where the presence of the interphase $q=1$ in $\overline{\Omega}$ poses some new difficulties to be tackled. The results proved in this work are the existence of $\lambda^* > 0$ such that no positive solutions are possible for $\lambda > \lambda^*$, the existence and structural properties of a branch of minimal solutions, $u_\lambda$, $0 < \lambda < \lambda^*$, and, finally, the existence for all $\lambda \in (0,\lambda^*)$ of a second positive solution.

#### Article information

Source
Topol. Methods Nonlinear Anal., Volume 47, Number 2 (2016), 613-639.

Dates
First available in Project Euclid: 13 July 2016

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1468413755

Digital Object Identifier
doi:10.12775/TMNA.2016.019

Mathematical Reviews number (MathSciNet)
MR3559923

Zentralblatt MATH identifier
1362.35116

#### Citation

García-Melián, Jorge; Rossi, Julio D.; de Lis, José C. Sabina. A variable exponent diffusion problem of concave-convex nature. Topol. Methods Nonlinear Anal. 47 (2016), no. 2, 613--639. doi:10.12775/TMNA.2016.019. https://projecteuclid.org/euclid.tmna/1468413755

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