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.