Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-24T03:44:20.017Z Has data issue: false hasContentIssue false
  • English
  • Français

On semilinear elliptic boundary value problems in unbounded domains

Published online by Cambridge University Press:  14 February 2012

Deb Kumar Bose
Deb Kumar Bose
Affiliation:
Institut für Mathematik, Ruhr–Universität, Bochum
Get access Rights & Permissions [Opens in a new window]

Synopsis

We consider the possibility of solving semilinear elliptic boundary value problems in unbounded domains. We first treat the case when the non-linear terms are independent of terms involving gradients. Using a monotone iteration scheme, we show that the existence of a weak subsolution v and a weak supersolution w ≧ v, implies the existence of a weak solution u, and v ≦ u ≦ w. We also state conditions which guarantee the existence of a solution when only a subsolution is known to exist. Next, we suppose the non-linear terms can depend on gradient terms. Using a method developed in [4], based on perturbation theory of maximal monotone operators, we prove the existence of a H2(Ω) solution lying between a given H2(Ω) subsolution v and a given H2(Ω) supersolution w ≧ v.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 77 , Issue 3-4 , 1977 , pp. 177 - 192
Copyright
Copyright © Royal Society of Edinburgh 1977

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Amann, H.. Nonlinear elliptic equations with nonlinear boundary conditions. Proc. 2nd Scheveningen Conference on Differential Equations 1975, to appear.CrossRefGoogle Scholar
2Hess, P.. Nonlinear perturbations of linear elliptic operators. Eds: Sleeman, B. D. and Michael, I. M.. Lecture Notes in Mathematics 415 (Berlin: Springer, 1974).Google Scholar
3Puel, J. P.. Controle optimal sur les solutions minimum ou maximum de certains problèmes semi Iinéaires, to appear.Google Scholar
4Brill, H.. On the solvability of semilinear elliptic equations with nonlinear boundary conditions. Math. Ann. 222 (1976), 37–48.CrossRefGoogle Scholar
5Deuel, J. and Hess, P.. A criterion for the existence of solutions of nonlinear elliptic boundary value problems. Proc. Roy. Soc. Edinburgh Sect. A 74 (1976), 49–54.CrossRefGoogle Scholar
6Hess, P.. On nonlinear elliptic problems in unbounded domains, to appear.Google Scholar
7Kusano, T.. On bounded solutions of exterior B.V.P.'s for linear and quasilinear elliptic differential equations. Japan J. Math. 35 (1965), 31–59.CrossRefGoogle Scholar
8Browder, F. E.. On the spectral theory of strongly elliptic differential operators. Math. Ann. 142 (1961), 22–130.CrossRefGoogle Scholar
9Edmunds, D. and Evans, W. D.. Elliptic and degenerate elliptic operators in unbounded domains. Ann. Scuola Norm. Sup. Pisa. 27 (1973), 591–640.Google Scholar
10Herve, R. M.. Un principle du maximum pour les sous-solutions locales d'une equation uniformément elliptique de la forme . Ann. Inst. Fourier 14 (1964), 493–508.CrossRefGoogle Scholar
11Krasnosel'skii, M. A.. Positive solutions of operator equations (Groningen: Noordhoff, 1964).Google Scholar
12Amann, H.. On the number of solutions of nonlinear equations in ordered Banach spaces. J. Functional Analysis 11 (1972), 346–384.CrossRefGoogle Scholar
13Vainberg, M. M.. Variational methods for the study of nonlinear operators (San Francisco: Holden Day, 1964).Google Scholar
14Morrey, C. B.. Multiple integrals in the calculus of variations. Die Grundlehren der Math. Wiss. 130 (Berlin: Springer, 1966).Google Scholar
15Brezis, H.. Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. North Holland Math. Studies 5 (Amsterdam: North Holland, 1973).Google Scholar
16Bartle, R. G.. The elements of integration (New York: Wiley, 1966).Google Scholar
17Mizohata, S.. The theory of partial differential equations (Edinburgh: Univ. Press, 1973).Google Scholar
18Lewy, H. and Stampacchia, G.. Regularity of the solution of a variational inequality. Comm. Pure Appl. Math. 22 (1969), 153–188.CrossRefGoogle Scholar