Electron. J. Diff. Equ., Vol. 2011 (2011), No. 88, pp. 1-12.

Asymptotic behavior of ground state solutions for sublinear and singular nonlinear Dirichlet problems

Rym Chemmam, Abdelwaheb Dhifli, Habib Maagli

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 $C_{loc}^{\alpha }(\mathbb{R}^n)$, $0<\alpha <1$, 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 5, 2011.
Math Subject Classifications: 31B05, 31C35, 34B27, 60J50.
Key Words: Asymptotic behavior; Dirichlet problem; ground sate solution; singular equations; sublinear equations.

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Rym Chemmam
Département de Mathématiques, Faculté des Sciences de Tunis
Université Tunis El Manar, Campus universitaire
2092 Tunis, Tunisia
email: Rym.Chemmam@fst.rnu.tn
Abdelwaheb Dhifli
Département de Mathématiques, Faculté des Sciences de Tunis
Université Tunis El Manar, Campus universitaire
2092 Tunis, Tunisia
email: dhifli_waheb@yahoo.fr
Habib Mâagli
Département de Mathématiques, Faculté des Sciences de Tunis
Université Tunis El Manar, Campus universitaire
2092 Tunis, Tunisia
email: habib.maagli@fst.rnu.tn

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