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Zbl 0536.35024
Kusano, Takaŝi; Swanson, Charles A.
Asymptotic properties of semilinear elliptic equations. (English)
[J] Funkc. Ekvacioj, Ser. Int. 26, 115-129 (1983). ISSN 0532-8721

Semilinear elliptic equations of the type (1) $\Delta u+f(x,u)=0$ are considered in exterior domains $\Omega \subset {\bbfR}\sp n$, $n\ge 2$, where $\Delta$ denotes the n-dimensional Laplacian and f is locally Hölder continuous in $\Omega \times {\bbfR}\sp+$. It is assumed that f(x,u) is majorized above and below by positive-valued functions of the type $\phi$ ($\vert x\vert,u)$ which are nonincreasing in u. Necessary and sufficient conditions are proved for the existence of positive solutions u(x) of(1) with specific asymptotic behavior as $\vert x\vert \to \infty$ in some exterior domain in ${\bbfR}\sp n$. In particular, sufficient conditions are given for the singular stationary Klein-Gordon equation $\Delta u-k\sp 2u+p(x)u\sp{-\lambda}=0,$ for positive constants k and $\lambda$, to have positive solutions $u\sb{\pm}(x)$ such that $\vert x\vert\sp{(n-1)/2}\exp(\pm k\vert x\vert)u\sb{\pm}(x)$ are bounded above and below by positive constants in some exterior domain.

MSC 2000:: *35J60 Nonlinear elliptic equations
35B40 Asymptotic behavior of solutions of PDE
35B05 General behavior of solutions of PDE
35J15 Second order elliptic equations, general
35A05 General existence and uniqueness theorems (PDE)

Keywords: semilinear elliptic equations; exterior domains; existence; positive solutions; singular stationary Klein-Gordon equation

Cited in: Zbl 0646.35030

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