## 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

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 $$\partial D$$.

### MSC:

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