## Asymptotic behavior of ground state solutions for sublinear and singular nonlinear Dirichlet problems.(English)Zbl 1225.31002

Summary: In this article, we are concerned with the asymptotic behavior of the classical solution to the semilinear boundary-value problem $\Delta u+a(x)u^{\sigma }=0$ in $$\mathbb{R}^n$$, $$u>0$$, $$\lim_{|x|\to \infty }u(x)=0$$, where $$\sigma <1$$. The special feature is to consider the function $$a$$ in $$C_{loc}^{\alpha }(\mathbb{R}^n)$$, $$0<\alpha <1$$, such that there exists $$c>0$$ satisfying $\frac{1}{c}\frac{L(|x| +1)}{(1+|x| )^{\lambda }} \leq a(x)\leq c\frac{L(|x| +1)}{(1+|x| )^{\lambda }}\,,$ where $L(t):=\exp \bigg(\int_1^t\frac{z(s)}{s}ds\bigg)$ with $$z\in C\big([1,\infty )\big)$$ such that $$\lim_{t\to \infty } z(t)=0$$. The comparable asymptotic rate of $$a(x)$$ determines the asymptotic behavior of the solution.

### MSC:

 31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions 31C35 Martin boundary theory 34B27 Green’s functions for ordinary differential equations 60J50 Boundary theory for Markov processes
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