Sur les solutions maximales de problèmes elliptiques nonlinéaires: Bornes isopérimétriques et comportement asymptotique. (Maximal solutions in nonlinear elliptic problems: Isoperimetric estimates and asymptotic behaviour). (French) Zbl 0726.35041

Bandle, Catherine; Marcus, Moshe

C. R. Acad. Sci., Paris, Sér. I 311, No. 2, 91-93 (1990).

Let D be a domain in \({\mathbb{R}}^ N\) with smooth boundary and consider the problem \[ (*)\quad \Delta u=f(u),\quad u\geq 0,\quad u\not\equiv 0\text{ in } D. \] The authors study the “largest” solution of (*), which can be defined via two different recipes. Writing S for the set of all solutions to (*), we define U by \(U(x)=\sup_{u\in S} u(x)\), and we define V by \(V(x)=\lim_{j\to \infty}v_ j(x)\), where \(v_ j\) solves \(\Delta v_ j=f(v_ j)\) in D, \(v_ j=j\) on \(\partial D\). Under suitable technical assumptions on f (e.g. \(f(0)=0\), f is differentiable and increasing, and the anti-derivative F given by \(F(t)=\int^{t}_{0}f(s)ds\) satisfies \(F^{-1/2}\) is integrable at infinity), the authors assert that \(U=V\) and they describe the behavior of V and \(\nabla V\) near \(\partial D\).

Reviewer: G.M.Lieberman (Ames)

MSC:

35J65	Nonlinear boundary value problems for linear elliptic equations
35B40	Asymptotic behavior of solutions to PDEs

Keywords:

isoperimetric estimates; semilinear elliptic; gradient; asymptotic behaviour; maximal solution