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.


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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