Existence and a priori estimates for semilinear elliptic systems of Hardy type (Q2921631)

The author considers semilinear elliptic systems of the form NEWLINE\[NEWLINE -\Delta u=a(x)| x| ^{-\kappa}v^q,\quad -\Delta v=b(x)| x| ^{-\lambda}u^p,\qquad x\in \Omega ,\tag{1} NEWLINE\]NEWLINE where \(\Omega \) is a smooth bounded domain in \(n\)-dimensional Euclidean space satisfying \(0\in \partial \Omega \), \(a,b\) are nonnegative and bounded, \(\kappa ,\lambda \in (0,2)\), \(p,q>0\), \(pq>1\). He finds sufficient conditions on \(\kappa ,\lambda ,p,q\) guaranteeing the existence, boundedness and a priori estimates of nonnegative very weak solutions of (1) complemented by the homogeneous Dirichlet boundary conditions. He also shows that his conditions are essentially optimal. The proofs of the boundedness and a priori estimates are based on a bootstrap argument in weighted Lebesgue spaces.