Boundary singular problems for quasilinear equations involving mixed reaction-diffusion

We study the existence of solutions to the problem

$$\begin{array}{rl}-\mathrm{\Delta }u+u^p-M|\mathrm{\nabla }u{|}^q=0& \text{in }\mathrm{\Omega },\\ u=\mu & \text{on }\partial \mathrm{\Omega }\end{array}$$

in a bounded domain $\mathrm{\Omega }$, where $p>1$, \(1, $M>0$, $\mu$ is a nonnegative Radon measure in $\partial \mathrm{\Omega }$, and the associated problem with a boundary isolated singularity at $a\in \partial \mathrm{\Omega },$

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

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 $$\mu (K)\le cmin\{ca{p}_{\frac{2}{p},p^{\prime }}^{\partial \mathrm{\Omega }},ca{p}_{\frac{2-q}{q},q^{\prime }}^{\partial \mathrm{\Omega }}\}\text{forall compacts }K\subset \partial \mathrm{\Omega }.$$ Problem [eng_A2] depends on several critical exponents on $p$ and $q$ as well as the position of $q$ with respect to ${\displaystyle \frac{2p}{p+1}}$.