Decaying solutions for semilinear elliptic equations in exterior domains. (English) Zbl 0956.35049

Stańczy, Robert

Topol. Methods Nonlinear Anal. 14, No. 2, 363-370 (1999).

The author considers the following problem \[ \begin{gathered} -\Delta u = f(\|x\|,u)\quad\text{for}\quad \|x\|\geq 1,\quad x\in\mathbb R^n, \quad n\geq 3\\ u(x) = 0 \quad\text{for}\quad \|x\|= 1,\quad \lim_{\|x\|\to\infty} u(x) = 0. \end{gathered} \] It is proved the existence of at least one radial solution. Typified nonlinearities: \(f(r,v) = h(r)g(v)\), where \(h: [1,\infty)\rightarrow\mathbb R\) and \(g: \mathbb R\rightarrow\mathbb R\) are continuous, and (a) \(|h(r)|\leq\text{const}\cdot r^\beta\), \(\beta<-2\); (b) \(g(0)\neq 0\); (c) \(\varlimsup_{|v|\to\infty}|g(v)|/|v|< C\), where \(C\) is an explicitly defined constant; or (\(\text{c}'\)) \(g(v)v< 0\) at large \(|v|\).

Reviewer: Serghey G.Suvorov (Donetsk)

MSC:

35J65	Nonlinear boundary value problems for linear elliptic equations
34B15	Nonlinear boundary value problems for ordinary differential equations
45G10	Other nonlinear integral equations
34B27	Green’s functions for ordinary differential equations

Keywords:

semilinear elliptic equations; exterior domain; Green function