Hostname: page-component-8448b6f56d-dnltx Total loading time: 0 Render date: 2024-04-23T12:12:32.748Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Dirichlet problem for weakly non-linear elliptic partial differential equations

Published online by Cambridge University Press:  14 February 2012

E. N. Dancer
E. N. Dancer
Affiliation:
Department of Mathematics, University of New England, Australia
Get access Rights & Permissions [Opens in a new window]

Synopsis

We study the existence of solutions of the Dirichlet problem for weakly nonlinear elliptic partial differential equations. We only consider cases where the nonlinearities do not depend on any partial derivatives. For these cases, we prove the existence of solutions for a wide variety of nonlinearities.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 76 , Issue 4 , 1977 , pp. 283 - 300
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

1Alexandroff, A. D.. Investigations of the maximum principle. Izv. Vyss. Učebn. Zaved. Matematika 5 (1958), 126–157.Google Scholar
2Ambrosetti, A. and Prodi, G.. On the inversion of some differentiable mappings with singularities between Banach spaces. Ann. Mat. Pura Appl. 93 (1972), 231–246.CrossRefGoogle Scholar
3Chicco, M.. Some properties of the first eigenvalue and first eigenfunction of linear second order elliptic partial differential equations in divergence form. Boll. Un. Mat. Ital. 5 (1972), 245–254.Google Scholar
4Dancer, E. N.. On a nonlinear elliptic boundary-value problem. Bull. Austral. Math. Soc. 12 (1975), 399–405.CrossRefGoogle Scholar
5Fučík, S.. Subjectivity of operators involving linear noninvertible part and nonlinear compact perturbation. Funkcial. Ekvac. 17 (1974), 73–83.Google Scholar
6Fučík, S.. Nonlinear equations with noninvertible linear part. Czechoslovak Math. J. 24 (1974), 467–495.CrossRefGoogle Scholar
7Fučík, S.. Further remark on a paper of E. M. Landesman and A. C. Lazer. Comment. Math. Univ. Carolinae 16 (1974), 259–271.Google Scholar
8Fučík, S.. Boundary value problems with jumping nonlinearities. Časopis Pěst. Mat. 101 (1976), 69–87.CrossRefGoogle Scholar
9Fučík, S., Kučera, M. and Nečas, J.. Ranges of nonlinear asymptotically linear operators. J. Differential Equations 17 (1975), 375–394.CrossRefGoogle Scholar
10Fučík, S. and Lovicar, V.. Boundary value and periodic problem for the equation x”(t)+g(x(t))= p(t). Comment. Math. Univ. Carolinae. 15 (1974), 351–355.Google Scholar
11Hess, P.. On semi-coercive nonlinear problems. Indiana Univ. Math. J. 23 (1974), 645–654.CrossRefGoogle Scholar
12Krasnosel'skii, M. A.. Positive solutions of operator equations (Groningen: Noordhoff, 1964).Google Scholar
13Krasnosel'skii, M. A. and Lifsic, E. A.. Duality principles for boundary-value problems. Soviet Math. Dokl. 8 (1967), 1236–1239.Google Scholar
14Ladyzhenskaya, O. A. and Ural'tseva, N.. Linear and quasilinear elliptic equations (New York: Academic Press, 1968).Google Scholar
15Landesman, E. M. and Lazer, A. C.. Nonlinear perturbations of linear elliptic equations at resonance. J. Math. Mech. 19 (1970), 609–623.Google Scholar
16Lazer, A. C.. On Schauder's fixed point theorem and forced second-order nonlinear oscillations. J. Math. Anal. Appl. 21 (1968), 421–425.CrossRefGoogle Scholar
17Lazer, A. C. and Leach, D. E.. Bounded perturbations of forced harmonic oscillators at resonance. Ann. Mat. Pura Appl. 82 (1970), 49–68.CrossRefGoogle Scholar
18Nečas, J.. On the range of nonlinear operators with asymptotes which are not invertible. Comment Math. Univ. Carolinae. 14 (1973), 63–72.Google Scholar
19Nirenberg, L.. Topics in nonlinear functional analysis (New York: Courant Inst., 1974).Google Scholar
20Protter, M. and Weinberger, H.. Maximum principles in differential equations (Englewood Cliffs: Prentice-Hall, 1967).Google Scholar
21Riesz, F. and Sz-Nagy, B.. Functional analysis (New York: Ungar, 1955).Google Scholar
22Schatzman, M.. Problèmes aux limites nonlinéaires semicoercifs. C.R. Hebd. Séanc. Acad. Sci. Paris Ser. A. 275 (1972), 1305–1308.Google Scholar
23Schwartz, J. T.. Nonlinear functional analysis (New York: Gordon and Breach, 1969).Google Scholar
24Trudinger, N. S.. Linear elliptic operators with measurable coefficients. Ann. Scuola Norm. Sup. Pisa 27 (1973), 265–308.Google Scholar
25Williams, S. A.. A sharp sufficient condition for solution of a nonlinear elliptic boundary-value problem. J. Differential Equations 8 (1970), 580–586. Added in proof: J. L. Kazdan and F. W. Warner. (Comm. Pure Appl. Math. 28 (1975), 567–597) have independently obtained a number of results closely related to ours. In many cases, their results and ours are complementary.CrossRefGoogle Scholar