# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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 ${ℝ}^{N}$ with smooth boundary and consider the problem

$\left(*\right)\phantom{\rule{1.em}{0ex}}{\Delta }u=f\left(u\right),\phantom{\rule{1.em}{0ex}}u\ge 0,\phantom{\rule{1.em}{0ex}}u¬\equiv 0\phantom{\rule{4.pt}{0ex}}\text{in}\phantom{\rule{4.pt}{0ex}}D·$

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 $U\left(x\right)={sup}_{u\in S}u\left(x\right)$, and we define V by $V\left(x\right)={lim}_{j\to \infty }{v}_{j}\left(x\right)$, where ${v}_{j}$ solves ${\Delta }{v}_{j}=f\left({v}_{j}\right)$ in D, ${v}_{j}=j$ on $\partial D$. Under suitable technical assumptions on f (e.g. $f\left(0\right)=0$, f is differentiable and increasing, and the anti-derivative F given by $F\left(t\right)={\int }_{0}^{t}f\left(s\right)ds$ 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 of PDE