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Zbl 1072.35071
Lin, Chang-Shou; Chen, Chiun-Chuan
A spherical Harnack inequality for singular solutions of nonlinear elliptic equations. (English)
[J] Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 30, No. 3-4, 713-738 (2001). ISSN 0391-173X

The authors continue their previous work [Duke Math. J. 78, No. 2, 315--334 (1995; Zbl 0839.35014)] on the local behaviour of positive singular solutions of the equation $$ \Delta u + g(u) = 0 \quad\text{in}\quad B_2\backslash \Gamma \leqno(*) $$ where $B_R$ denotes the open ball in $\bbfR^n$, $n\ge 3$, of radius $R$ centred at the origin, $g$ is a positive, locally bounded function on $(0,\infty)$ with $t^{-{{n+2}\over{n-2}}}g(t)$ nonincreasing in $t$ for large $t$, and $\Gamma$ is a closed subset of $\bar B_1$ with vanishing Newton capacity. Unlike many previous works in which $\Gamma$ was assumed to be discrete, this assumption allows $\Gamma$ to be as large as a codimension two submanifold of $\bar B_1$. The main result is the following spherical Harnack inequality: $$ {{g(u(x))}\over{u(x)}} \le c\,d(x)^{-2} \quad\text{for}\quad |x|\le 1 $$ where $c$ is a positive constant and $d(x)=dist(x,\Gamma)$. This result is new even when $\Gamma=\{0\}$ and $u$ is radially symmetric. \par A corollary of the main theorem is that if $u$ is a positive solution of $(*)$ with $\Gamma=\{0\}$ and $0$ is a nonremovable singularity, then $\lim_{x\rightarrow 0} u(x)=+\infty$ and $u$ is asymptotically symmetric as $x\rightarrow 0$ in the sense that $u(x)=\bar u(r)(1+o(1))$ for $r=|x|$ where $\bar u(r)$ is the average of $u$ over the sphere $S_r=\{x:|x|=r\}$ and $o(1)\rightarrow 0$ as $r\rightarrow 0$. \par The proof uses the method of moving planes and a localization technique of {\it R. Schoen} [Commun. Pure Appl. Math. 41, No. 3, 317--392 (1988; Zbl 0674.35027)].
[John Urbas (Canberra)]

MSC 2000:: *35J60 Nonlinear elliptic equations
35B45 A priori estimates
35B33 Critical exponents

Citations: Zbl 0839.35014; Zbl 0674.35027

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