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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 \(f\).

MSC:

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