Boundary behavior for large solutions to elliptic equations with singular weights (Q881615)

The author deals with the following semilinear elliptic boundary value problem \[ \begin{gathered} \Delta_x u= a(x) f(u)\quad\text{in }\Omega,\\ u=+\infty\quad\text{on } \partial\Omega,\end{gathered}\tag{1} \] where \(\Omega\) is a bounded \(C^{2,\gamma}\) domain of \(\mathbb{R}^d\), \(a(x)\) is a positive weight function and \(f(s)\) is a \(C^1\) function defined in \([0,+\infty)\). The boundary condition has to be unbounded as \(u(x)\to+\infty\) when \(r(x):= \text{dist}(x,\partial\Omega)\to 0+\). In (1) the weight function \(a(x)\) is assumed to be singular near the boundary of \(\Omega\). The author is mainly interested in the existence and uniqueness of positive solutions for (1).