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Uniqueness and asymptotic behavior of solutions with boundary blow-up for a class of nonlinear elliptic equations. (English) Zbl 0877.35042

The authors study properties of positive solutions of

${\Delta }u+hu-k{u}^{p}=f\phantom{\rule{2.em}{0ex}}\left(1\right)$

in a (possibly) nonsmooth $N$-dimensional domain ${\Omega }$, $N\ge 2$, subject to the condition

$u\left(x\right)\to \infty \phantom{\rule{1.em}{0ex}}\text{if}\phantom{\rule{1.em}{0ex}}\delta \left(x\right):=\text{dist}\left(x,\partial {\Omega }\right)\to 0·\phantom{\rule{2.em}{0ex}}\left(2\right)$

Here $p>1$ and $h$, $k$, $f$ are continuous in $\overline{{\Omega }}$ with $k>0$ and $f\ge 0$. Positive solutions of (1) satisfying (2) are called large solutions. A central point of this paper is the following localization principle: let ${\Omega }$ be a (not necessarily bounded) domain having the graph property and suppose $u$ is a positive solution of (1) satisfying $u\left(x\right)\to \infty$ locally uniformly as $x\to {\Gamma }$, where ${\Gamma }\subset \partial {\Omega }$ is relatively open. If $v$ is a large solution, then $v\left(x\right)/u\left(x\right)\to 1$ locally uniformly as $x\to {\Gamma }$. Closely related to this is a uniqueness result for large solutions in bounded domains having the graph property. For bounded Lipschitz domains the authors prove the existence of positive constants ${c}_{1}\le {c}_{2}$ such that the (unique) large solution $u$ of (1) satisfies ${c}_{1}\delta {\left(x\right)}^{-\frac{2}{p-1}}\le u\left(x\right)\le {c}_{2}\delta {\left(x\right)}^{-\frac{2}{p-1}}$ for all $x\in {\Omega }$. This is also a consequence of the localization principle and an existence theorem, obtained for large solutions in bounded domains satisfying the exterior cone condition.

If the domain is not Lipschitz, the rate of blow-up at the boundary may be lower. This is proved for domains having a re-entrant cusp in the case $p\ge \left(N-1\right)/\left(N-3\right)$. Finally, the authors discuss the dependence of large solutions on the function $k$ and the domain ${\Omega }$.

##### MSC:
 35J60 Nonlinear elliptic equations 35J67 Boundary values of solutions of elliptic equations 35B40 Asymptotic behavior of solutions of PDE