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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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