Existence and uniqueness theorems for singular anisotropic quasilinear elliptic boundary value problems

Abstract:

On bounded domains $\Omega \subset \mathbb {R}^2$ we consider the anisotropic problems $u^{-a}u_{xx}+u^{-b}u_{yy}=p(x,y)$ in $\Omega$ with $a,b>1$ and $u=\infty$ on $\partial \Omega$ and $u^cu_{xx}+u^du_{yy}+q(x,y)=0$ in $\Omega$ with $c,d\geq 0$ and $u=0$ on $\partial \Omega$. Moreover, we generalize these boundary value problems to space-dimensions $n>2$. Under geometric conditions on $\Omega$ and monotonicity assumption on $0<p,q\in \mathcal {C}^\alpha (\overline {\Omega })$ we prove existence and uniqueness of positive solutions.