Existence and asymptotic properties of singular solutions of nonlinear elliptic equations in R n ∖ { 0 } $R^{n}\backslash\{0\}$

Abstract

Abstract We consider the following singular semilinear problem { Δ u ( x ) + p ( x ) u γ = 0 , x ∈ D ( in the distributional sense ) , u > 0 , in D , lim | x | → 0 | x | n − 2 u ( x ) = 0 , lim | x | → ∞ u ( x ) = 0 , $$ \textstyle\begin{cases} \Delta u(x)+p(x)u^{\gamma }=0,\quad x\in D ~(\text{in the distributional sense}), \\ u>0,\quad \text{in }D, \\ \lim_{ \vert x \vert \rightarrow 0} \vert x \vert ^{n-2}u(x)=0, \\ \lim_{ \vert x \vert \rightarrow \infty }u(x)=0,\end{cases} $$ where γ < 1 $\gamma <1$ , D = R n ∖ { 0 } $D=\mathbb{R}^{n}\backslash \{0\}$ ( n ≥ 3 $n\geq 3$ ) and p is a positive continuous function in D, which may be singular at x = 0 $x=0$ . Under sufficient conditions for the weighted function p ( x ) $p(x)$ , we prove the existence of a positive continuous solution on D, which could blow-up at the origin. The global asymptotic behavior of this solution is also obtained.

Keywords

• Slowly varying functions
• Kato class
• Asymptotic properties
• Blow-up