×

zbMATH — the first resource for mathematics

On some quasilinear elliptic boundary value problems with conditions at infinity. (English) Zbl 0543.35037
L’A. considère le problème \(Au=p(x,u,\nabla u)\) sur \(\Omega\), \(u=0\) sur \(\partial \Omega\) où \(\Omega \subset {\mathbb{R}}^ N\) est un domaine extérieur, A est un opérateur linéaire uniformément elliptique sous forme divergente et p(x,u,\(\eta)\) est à croissance sous- quadratique par rapport à \(\eta\). L’A. prouve l’existence d’une solution sous l’hypothèse qu’il existe une sous-solution \(\phi\) et une sur-solution \(\psi\) telles que \(\phi \leq 0\leq \psi\). La démonstration utilise des espaces de Sobolev avec poids. Il généralise un résultat de V. Benci et D. Fortunato [Nonlinear Anal., Theory, Methods, Appl. 3, 157-174 (1979; Zbl 0404.35043)].
Reviewer: H.Brézis

MSC:
35J65 Nonlinear boundary value problems for linear elliptic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Benci, V; Fortunato, D, Some nonlinear elliptic problems with asymptotic conditions, Nonlinear anal., 3, 157-174, (1979) · Zbl 0404.35043
[2] Bérestycki, H; Lions, P.L, Some applications of the method of super and subsolutions, (), 16-41
[3] Brézis, H; Stampacchia, G, Sur la régularité de la solution d’inéquations elliptiques, Bull. soc. math. France, 96, 153-180, (1968) · Zbl 0165.45601
[4] Càc, N.P, Nonlinear elliptic boundary value problems for unbounded domains, J. differential equations, 45, 191-198, (1982) · Zbl 0508.35032
[5] Hess, P, Nonlinear elliptic problems in unbounded domains, () · Zbl 0367.35021
[6] Il’in, A.M; Kalashnikov, A.S; Oleinik, O.A, Linear equations of the second order of parabolic type, Russian math. surveys, 17, 3, 1-143, (1962) · Zbl 0988.35001
[7] Ladyzhenskaya, O.A; Ural’tseva, N.N, Linear and quasilinear elliptic equations, (1968), Academic Press New York · Zbl 0164.13002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.