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Asymptotic behavior of solutions to \(\Delta u+Ku^{\sigma}=0\) on \(R^ n\) for n\(\geq 3\). (English) Zbl 0705.35012

The elliptic equation \(-\Delta u=Ku^{\sigma}\) is under consideration in \({\mathbb{R}}^ n\), where \(n\geq 3\), \(\sigma >0\), and K: \({\mathbb{R}}^ n\to {\mathbb{R}}\) is a nontrivial locally Hölder continuous function with \(K(x)=O(| x|^{-a})\) as \(| x| \to \infty\) for some constant \(a>2\). If u(x) is a bounded positive solution, the first theorem establishes the existence of a constant \(u_{\infty}\) such that \(u(x)=u_{\infty}+O(| x|^ b)\) as \(| x| \to \infty\) for all \(b>\max \{2-a,2-n\}\). If \(a>n\), a more detailed asymptotic description is given. If \(\sigma >1\) and K(x)\(\leq 0\) in \({\mathbb{R}}^ n\), then for any prescribed \(u_{\infty}>0\), there exists a unique positive solution u such that \(u(x)=u_{\infty}+O(| x|^ b)\) at \(\infty\) for all b as above.

MSC:

35B40 Asymptotic behavior of solutions to PDEs
35J60 Nonlinear elliptic equations
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