Boundary Singular Problems for Quasilinear Equations Involving Mixed Reaction–Diffusion

Abstract

We study the existence of solutions to the problem

$\begin{array}{c}\begin{array}{cccc}-\Delta u+{u}^{p}-M{\left|\nabla u\right|}^{q}=0& \text{in}&\Omega ,& (1)\end{array}\\ \begin{array}{ccc}u=\mu & \text{on}& \partial\Omega \end{array}\end{array}$

in a bounded domain Ω, where p > 1, 1 < q < 2, M > 0, μ is a nonnegative Radon measure in ∂Ω, and the associated problem with a boundary isolated singularity at a ∈ ∂Ω,

$\begin{array}{c}\begin{array}{cccc}-\Delta u+{u}^{p}-M{\left|\nabla u\right|}^{q}=0& \text{in}&\Omega ,& (2)\end{array}\\ \begin{array}{ccc}u=0& \text{on}& \partial\Omega \end{array}\backslash \left\{\alpha \right\}.\end{array}$

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

$\begin{array}{cc}\mu \left(K\right)\le c\text{min}\left\{{cap}_{\frac{2}{p},{p}{\prime}}^{\partial\Omega },{cap}_{\frac{2-q}{q},{q}{\prime}}^{\partial\Omega }\right\}& \text{for\;all\;compacts\;}K\subset \partial\Omega .\end{array}$

Problem (2) depends on several critical exponents on p and q as well as the position of q with respect to $\frac{2p}{p+1}$.