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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 N with smooth boundary and consider the problem


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 uS u(x), and we define V by V(x)=lim j v j (x), where v j solves Δv j =f(v j ) in D, v j =j on 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)= 0 t 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 V near D.

35J65Nonlinear boundary value problems for linear elliptic equations
35B40Asymptotic behavior of solutions of PDE