×

Uniqueness and boundary behavior of large solutions to elliptic problems with singular weights. (English) Zbl 1174.35386

In this paper the authors study the problems corresponding of power and exponential type, that is \[ \begin{cases} \Delta u=a(x)u^m\quad & \text{in }\Omega\\ u=+\infty & \text{on }\partial\Omega\end{cases}\tag{1} \] and \[ \begin{cases} \Delta v=a(x)e^v\quad & \text{in }\Omega\\ v=+\infty & \text{on }\partial\Omega,\end{cases}\tag{2} \] where the weight function \(a(x)\) is assumed to be Hölder continuous, growing like a negative power of \(d(x)=\text{dist}(x,\partial\Omega)\) near \(\partial\Omega\). The authors prove existence and nonexistence results, uniqueness and asymptotic estimates near the boundary for both the solutions and their normal derivatives.

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
35B45 A priori estimates in context of PDEs
PDFBibTeX XMLCite
Full Text: DOI