Asymptotic behavior of ground state solutions for sublinear and singular nonlinear Dirichlet problems - Rym Chemmam; Abdelwaheb Dhifli; Habib Maagli (Electronic Journal of Differential Equations, 2011)
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 $mathbbR^n$, $u>0$, $lim_{|x|o infty }u(x)=0$, where $sigma <1$. The special feature is to consider the function $a$ in $C_{m loc}^{alpha }(mathbbR^n)$, $0<alpha <1$, such that there exists $c>0$ satisfying $$ frac{1}{c}frac{L(|x| +1)}{(1+|x| )^{lambda }} leq a(x)leq cfrac{L(|x| +1)}{(1+|x| )^{lambda }}, $$ where $L(t):=exp ig(int_1^tfrac{z(s)}{s}dsig)$, with $zin C([1,infty ))$ such that $lim_{to infty } z(t)=0$. The comparable asymptotic rate of $a(x)$ determines the asymptotic behavior of the solution.