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Boundary singularities of solutions of some nonlinear elliptic equations. (English) Zbl 0766.35015
The authors study the boundary singularities of the solutions of a semilinear elliptic equation of (#) $$-\Delta u+g(u)=0$$ in $$G$$. They assume $$G$$ is a bounded open set in $$\mathbb{R}^ N$$ with a $$C^ 2$$ boundary $$\partial G$$.
They prove the following three results. Assume $$g$$ is continuous on $$\mathbb{R}$$ and satisfies $\liminf_{r\to\infty} g(r)r^{-(N+1)/(N-1)}>0, \qquad \limsup_{r\to-\infty} g(r)|-r|^{-(N+1)/(N-1)}<0,$ $$\varphi$$ is continuous on $$\partial G$$, $$F$$ is a discrete subset of $$\partial G$$, and $$u\in C^ 2(G)\cap C(\overline{G}\setminus F)$$ is a solution of (#) such that $$u=\varphi$$ on $$\partial G\setminus F$$. Then $$u$$ is the restriction to $$\overline{G}\setminus F$$ of a $$C(\overline{G})$$-function $$\overline{u}$$ satisfying (#) and $$\overline{u}=\varphi$$ on $$\partial G$$.
Let $$\rho(x)$$ be the distance between $$x$$ and $$\partial G$$ and let $$g$$ be a continuous nondecreasing, real-valued function satisfying $$\int_ 1^ \infty (g(s)+| g(-s)|)s^{-2N/(N-1)} ds<\infty$$. Then for any measure $$\mu$$ on $$\partial G$$ there exists a unique solution $$u\in L^ 1(G)$$ such that $$\rho(.)g(u)\in L^ 1(G)$$ satisfying $$\int_ G\{- u\Delta\zeta+g(u)\zeta\} dx=-\langle \mu,\partial\zeta/\partial\nu\rangle$$ for any $$\zeta$$ belonging to the space $$C^{1,1}(\overline{G})\cap W_ 0^{1,\infty}(G)$$ of $$C^ 1$$ functions with Lipschitz continuous gradient, vanishing on $$\partial G$$.
Assume $$1<q<(N+1)/(N-1)$$ and let $$u^ e$$ be the extension of $$u$$ by 0 outside $$\overline{G}$$. Then there exists $$\omega\in C^ 2(\overline{G})$$ such that $$(-\Delta_ S N-1)\omega+\omega| \omega|^{q-1}=2(q-1)^{-1} \{2q(q-1)^{-1}-N\}\omega$$ in $$S_ +^{N-1}=S^{N-1}\cap\{x_ N>0\}$$ with $$\omega=0$$ on $$\partial S_ +^{N-1}$$, and $$\lim_{r\to 0} r^{2/(q-1)} u^ e(r,\sigma)=\omega(\sigma)$$ uniformly on $$S_ +^{N-1}$$ when one of the three following conditions is fulfilled; (i) $$N=2$$, (ii) $$(N+2)/N\leq(N+1)/(N-1)$$, or (iii) $$u$$ is bounded below.

##### MSC:
 35J65 Nonlinear boundary value problems for linear elliptic equations 35J25 Boundary value problems for second-order elliptic equations 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
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