Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On the solvability of a nonlinear
second-order elliptic equation at resonance

Author: Chung-Cheng Kuo
Journal: Proc. Amer. Math. Soc. 124 (1996), 83-87
MSC (1991): Primary 35J65, 47H11, 47H15
MathSciNet review: 1301035
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We study the existence of solutions of the Neumann problem for semilinear second-order elliptic equations at resonance in which the nonlinear terms may grow superlinearly in one of the directions $u\to\infty$ and $u\to-\infty$, and sublinearly in the other. Solvability results are obtained under assumptions either with or without a Landesman-Lazer condition. The proofs are based on degree-theoretic arguments.

References [Enhancements On Off] (What's this?)

  • 1 S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions I, Comm. Pure Appl. Math. 12 (1959), 623--727. MR 23:A2610
  • 2 H. Berestycki and D. G. de Figueiredo, Double resonance in semilinear elliptic equations, Comm. Partial Differential Equations. 6 (1981), 91--22. MR 82f:35078
  • 3 C. P. Gupta, Perturbation of second order elliptic problems by unbounded nonlinearities, Nonlinear Anal. 6 (1982), 919--933. MR 84c:35042
  • 4 N. Hirano, Nonlinear perturbations of second order linear elliptic boundary value problems, Houston J. Math. 14 (1988), 105--114. MR 90c:35091
  • 5 R. Iannacci, M. N. Nkashama, and J. R. Ward, Jr., Nonlinear second order elliptic partial differential equations at resonance, Trans. Amer. Math. Soc. 311 (1989), 711--726. MR 90a:35091
  • 6 R. Kannan and R. Ortega, Landesman-Lazer conditions for problems with ``one-side unbounded'' nonlinearities, Nonlinear Anal.9 (1985), 1313--1317. MR 87d:35051
  • 7 E. M. Landesman and A. C. Lazer, Nonlinear perturbations of linear elliptic boundary value problems at resonance, J. Math. Mech. 19 (1970), 609--623. MR 42:2171
  • 8 N. G. Lloyd, Degree theory, Cambridge Univ. Press, Cambridge, 1978. MR 58:12558
  • 9 J. R. Ward, Jr., Perturbations with some superlinear growth for a class of second order elliptic boundary value problems, Nonlinear Anal. 6 (1982), 367--374. MR 83i:35076

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 35J65, 47H11, 47H15

Retrieve articles in all journals with MSC (1991): 35J65, 47H11, 47H15

Additional Information

Keywords: Second-order elliptic equation, Landesman-Lazer condition, Leray-Schauder degree
Received by editor(s): June 8, 1994
Additional Notes: This research was supported in part by the National Science Council of the Republic of China
Communicated by: Jeffrey Rauch
Article copyright: © Copyright 1996 American Mathematical Society

American Mathematical Society