A remark on the existence of large solutions via sub and supersolutions. (English) Zbl 1040.35026

The author considers the boundary blow-up elliptic problem \[ \begin{cases} \Delta u= a(x)f(u)\quad &\text{in }\Omega,\\ u=+\infty\quad &\text{on }\partial\Omega,\end{cases}\tag{1} \] where \(\Omega\subset\mathbb{R}^N\), \(N\geq 2\) is a smooth bounded domain, \(a(x)\) is a Hölder continuous positive function defined in \(\Omega\) and \(f\) is locally Hölder in \((0,+\infty)\). The author is mainly interested in the existence of positive classical solutions to (1), that is solutions \(u\in C^2(\Omega)\) to \(\Delta u= a(x)f(u)\) such that \(u(x)\to +\infty\) as \(d(x):= \text{dist}(x,\partial\Omega)\to 0\). Using sub- and supersolution techniques the author proves the existence of at least one positive solution under some suitable assumptions on \(a\) and \(f\).


35J60 Nonlinear elliptic equations
35J25 Boundary value problems for second-order elliptic equations
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