Boundary singular problems for quasilinear equations involving mixed reaction-diffusion

Abstract

We study the existence of solutions to the problem \[\label{eng_A1} \begin{array}{rl} -\Delta u+u^p-M|\nabla u|^q=0 \text{in }\;\Omega,\\ u=\mu \text{on }\;\partial\Omega \end{array}\] in a bounded domain \(\Omega\), where \(p1\), \(1q2\), \(M0\), \(\mu\) is a nonnegative Radon measure in \(\partial\Omega\), and the associated problem with a boundary isolated singularity at \(a\in\partial\Omega,\) \[\label{eng_A2} \begin{array}{rl} -\Delta u+u^p-M|\nabla u|^q=0 \text{in }\;\Omega,\\ u=0 \text{on }\;\partial\Omega\setminus\{a\}. \end{array}\] The difficulty lies in the opposition between the two nonlinear terms which are not on the same nature. Existence of solutions to[eng_A1] is obtained under a capacitary condition \[\mu(K)\leq c\min\left\{cap^{\partial\Omega}_{\frac{2}{p},p'},cap^{\partial\Omega}_{\frac{2-q}{q},q'}\right\}\quad\text{for all compacts }K\subset\partial\Omega.\] Problem[eng_A2] depends on several critical exponents on \(p\) and \(q\) as well as the position of \(q\) with respect to \(\dfrac{2p}{p+1}\).