Boundary singular problems for quasilinear equations involving mixed reaction-diffusion

L. Véron

Contemporary Mathematics. Fundamental Directions, Differential and functional differential equations, Tome 68 (2022) no. 4, pp. 564-574

Résumé

We study the existence of solutions to the problem \begin{equation} \begin{array}{rl} -\Delta u+u^p-M|\nabla u|^q=0 \text{in } \Omega,\\ u=\mu \text{on } \partial\Omega \end{array} \end{equation} in a bounded domain $\Omega$, where $p>1$, $1$, $M>0$, $\mu$ is a nonnegative Radon measure in $\partial\Omega$, and the associated problem with a boundary isolated singularity at $a\in\partial\Omega,$\begin{equation} \begin{array}{rl} -\Delta u+u^p-M|\nabla u|^q=0 \text{in } \Omega,\\ u=0 \text{on } \partial\Omega\setminus\{a\}. \end{array} \end{equation} The difficulty lies in the opposition between the two nonlinear terms which are not on the same nature. Existence of solutions to (1) is obtained under a capacitary condition $$ \mu(K)\leq c\min\left\{cap^{\partial\Omega}_{\frac{2}{p},p'},cap^{\partial\Omega}_{\frac{2-q}{q},q'}\right\} \text{for all compacts }K\subset\partial\Omega. $$ Problem (2) depends on several critical exponents on $p$ and $q$ as well as the position of $q$ with respect to $\dfrac{2p}{p+1}$.

Détail

Mots-clés : reaction-diffusion equation
Keywords: boundary singular problem, measure as boundary data, isolated boundary singularity.