Electronic Journal of Differential Equations,
Vol. 2011 (2011), No. 88, pp. 1-12.
Title: Asymptotic behavior of ground state solutions for sublinear and
singular nonlinear Dirichlet problems
Authors: Rym Chemmam (Univ. Tunis El Manar, Tunisia)
Abdelwaheb Dhifli (Univ. Tunis El Manar, Tunisia)
Habib Maagli (Univ. Tunis El Manar, Tunisia)
Abstract:
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 $C_{\rm loc}^{\alpha }(\mathbb{R}^n)$,
$0<\alpha <1$, 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 $L(t):=\exp \big(\int_1^t\frac{z(s)}{s}ds\big)$,
with $z\in C([1,\infty ))$ such that $\lim_{t\to \infty } z(t)=0$.
The comparable asymptotic rate of $a(x)$ determines the asymptotic
behavior of the solution.
Submitted April 13, 2011. Published July 05, 2011.
Math Subject Classifications: 31B05, 31C35, 34B27, 60J50.
Key Words: Asymptotic behavior; Dirichlet problem; ground sate solution;
singular equations; sublinear equations.