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A remark on the existence of explosive solutions for a class of semilinear elliptic equations. (English) Zbl 0964.35053
There has been considerable research on the problem of finding a positive solution $u$ of the differential equation ${\Delta }u=k\left(x\right)f\left(u\right)$ in ${\Omega }$, a bounded domain in ${ℝ}^{n}$, which diverges to infinity at $\partial {\Omega }$ when the function $k$ is a positive constant and $f\left(u\right)={e}^{u}$ or $f\left(u\right)={u}^{p}$ for some constant $p>1$. In this work, the author shows that this problem has a solution for a class of nonconstant functions $k$. The main conditions are that $k$ is uniformly bounded from below in ${\Omega }$ and does not blow up too fast near $\partial {\Omega }$. Specifically either $k\left(x\right)=O\left(d{\left(x\right)}^{\beta -2}\right)$ for some $\beta \in \left(0,2\right]$, where $d$ is distance to the boundary, or $d\in {L}^{q}$ for some $q>n/2$. The key step is to approximate the domain ${\Omega }$ from within by a sequence of smooth domains (so that $k$ is uniformly bounded on the approximating domains). The maximum principle shows that the sequence of solutions of the approximating problems with infinite boundary data (which are known to exist) is decreasing and a simple change of dependent variable along with the maximum principle shows that this sequence is bounded from below by a strictly positive function.
##### MSC:
 35J65 Nonlinear boundary value problems for linear elliptic equations 35B05 Oscillation, zeros of solutions, mean value theorems, etc. (PDE) 35A05 General existence and uniqueness theorems (PDE) (MSC2000) 35J25 Second order elliptic equations, boundary value problems