Decaying solutions for semilinear elliptic equations in exterior domains
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 \(|v|\).
scientific article; zbMATH DE number 1507588