Boundary singularities of solutions of some nonlinear elliptic equations. (English) Zbl 0766.35015

Gmira, Abdelilah; Véron, Laurent

Duke Math. J. 64, No. 2, 271-324 (1991).

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.

Reviewer: K.Kajitani (Ibaraki)

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

Keywords:

Dirichlet problem; boundary singularities; semilinear elliptic equation

References:

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