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Zbl 0726.35041
Bandle, Catherine; Marcus, Moshe
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)
[J] C. R. Acad. Sci., Paris, Sér. I 311, No.2, 91-93 (1990). ISSN 0764-4442

Let D be a domain in ${\bbfR}\sp N$ with smooth boundary and consider the problem $$(*)\quad \Delta u=f(u),\quad u\ge 0,\quad u\not\equiv 0\text{ in } D.$$ 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 $U(x)=\sup\sb{u\in S} u(x)$, and we define V by $V(x)=\lim\sb{j\to \infty}v\sb j(x)$, where $v\sb j$ solves $\Delta v\sb j=f(v\sb j)$ in D, $v\sb j=j$ 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 $F(t)=\int\sp{t}\sb{0}f(s)ds$ satisfies $F\sp{-1/2}$ is integrable at infinity), the authors assert that $U=V$ and they describe the behavior of V and $\nabla V$ near $\partial D$.
[G.M.Lieberman (Ames)]
MSC 2000:
*35J65 (Nonlinear) BVP for (non)linear elliptic equations
35B40 Asymptotic behavior of solutions of PDE

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

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