A variable exponent diffusion problem of concave-convex nature

Jorge García-Melián, J. D. Rossi, José C. Sabina de Lis

DOI: http://dx.doi.org/10.12775/TMNA.2016.019


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 $\Omega\subset \R^N$ is a bounded smooth domain, $\lambda\gt 0$ 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 $\lambda^* \gt 0$ such that no positive solutions are possible for $\lambda \gt \lambda^*$, the existence and structural properties of a branch of minimal solutions, $u_\lambda$, $0 \lt \lambda \lt \lambda^*$, and, finally, the existence for all $ \lambda \in (0,\lambda^*)$ of a second positive solution.


Variable exponent; concave-convex; minimal solution; a priori bounds; Leray-Schauder degree

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