## 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

### Keywords:

semilinear; bounded positive solution
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