Boundary singular problems for quasilinear equations involving mixed reaction-diffusion

We study the existence of solutions to 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 nonnegative Radon measure \(\partial\Omega\), and associated problem with boundary isolated singularity at \(a\in\partial\Omega,\)
 \[\label{eng_A2}
 u=0 }\;\partial\Omega\setminus\{a\}.
 The difficulty lies opposition between two nonlinear terms which are not on same nature. Existence to[eng_A1] obtained under 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 several critical exponents \(p\) \(q\) as well position respect \(\dfrac{2p}{p+1}\).