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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 $$\cases \Delta u= a(x)f(u)\quad &\text{in }\Omega,\\ u=+\infty\quad &\text{on }\partial\Omega,\endcases\tag1$$ where $\Omega\subset\bbfR^N$, $N\ge 2$ 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 $f$.

35J60Nonlinear elliptic equations
35J25Second order elliptic equations, boundary value problems
Full Text: EMIS