Boundary singular problems for quasilinear equations involving mixed reaction-diffusion

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 \(p>1\), \(1, \(M>0\), \(\mu\) is a nonnegative Radon measure in \(\partial\Omega\), and the associated problem with a boundary isolated singularity at \(a\in\partial\Omega,\)

\[\label{eng_A2}
\begin{array}{rl}
-\Delta u+u^p-M|\nabla u|^q=0 & \text{in }\;\Omega,\\
u=0 & \text{on }\;\partial\Omega\setminus\{a\}.
\end{array}\]

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 \(\dfrac{2p}{p+1}\).