On a singular elliptic boundary value problem in a ball. (English) Zbl 0707.35058
This paper deals with the existence of classical positive solutions to the elliptic boundary value problem
\[
\Delta u+f(x,u)=0,\quad x\in B=\{x\in {\mathbb{R}}^ N | \quad | x| <1\},\quad u=0\text{ on } \partial B,
\]
where \(\Delta\) is the N dimensional Laplacian, \(N\geq 2\), and f(x,u) is a positive, smooth function defined on \(B\times (0,\infty)\) and nonincreasing in \(u>0\). No smoothness is imposed on \(\partial (B\times (0,\infty))\) so that certain singularities are allowed. Additional assumptions on f are given which are sufficient to guarantee the existence of a positive solution \(u\in C^ 2(B)\cap C(\bar B)\).
Reviewer: R.Guenther
MSC:
35J60 | Nonlinear elliptic equations |
35A05 | General existence and uniqueness theorems (PDE) (MSC2000) |
35B05 | Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs |
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