Nonlinear perturbations of linear elliptic systems at resonance

Author:
Philip Korman

Journal:
Proc. Amer. Math. Soc. **140** (2012), 2447-2451

MSC (2010):
Primary 35J60

Published electronically:
November 21, 2011

MathSciNet review:
2898707

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We consider a semilinear system

whose linear part is at resonance. Here and the functions and are bounded and continuous. Assuming that for all , , and that the first harmonics of and lie on a certain straight line, we prove the existence of solutions. This extends a similar result for one equation, due to D.G. de Figueiredo and W.-M. Ni.

**1.**A. Ambrosetti and G. Prodi,*On the inversion of some differentiable mappings with singularities between Banach spaces*, Ann. Mat. Pura Appl. (4)**93**(1972), 231–246. MR**0320844****2.**Antonio Ambrosetti and Giovanni Prodi,*A primer of nonlinear analysis*, Cambridge Studies in Advanced Mathematics, vol. 34, Cambridge University Press, Cambridge, 1993. MR**1225101****3.**M. S. Berger and E. Podolak,*On the solutions of a nonlinear Dirichlet problem*, Indiana Univ. Math. J.**24**(1974/75), 837–846. MR**0377274****4.**Djairo G. de Figueiredo,*Semilinear elliptic systems: existence, multiplicity, symmetry of solutions*, Handbook of differential equations: stationary partial differential equations. Vol. V, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2008, pp. 1–48. MR**2497896**, 10.1016/S1874-5733(08)80008-3**5.**Djairo Guedes de Figueiredo and Wei Ming Ni,*Perturbations of second order linear elliptic problems by nonlinearities without Landesman-Lazer condition*, Nonlinear Anal.**3**(1979), no. 5, 629–634. MR**541873**, 10.1016/0362-546X(79)90091-9**6.**Philip Korman,*Curves of equiharmonic solutions, and ranges of nonlinear equations*, Adv. Differential Equations**14**(2009), no. 9-10, 963–984. MR**2548284****7.**E. M. Landesman and A. C. Lazer,*Nonlinear perturbations of linear elliptic boundary value problems at resonance*, J. Math. Mech.**19**(1969/1970), 609–623. MR**0267269****8.**Louis Nirenberg,*Topics in nonlinear functional analysis*, Courant Lecture Notes in Mathematics, vol. 6, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2001. Chapter 6 by E. Zehnder; Notes by R. A. Artino; Revised reprint of the 1974 original. MR**1850453****9.**Bernhard Ruf,*Superlinear elliptic equations and systems*, Handbook of differential equations: stationary partial differential equations. Vol. V, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2008, pp. 211–276. MR**2497908**, 10.1016/S1874-5733(08)80010-1

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2010):
35J60

Retrieve articles in all journals with MSC (2010): 35J60

Additional Information

**Philip Korman**

Affiliation:
Department of Mathematical Sciences, University of Cincinnati, Cincinnati, Ohio 45221-0025

Email:
kormanp@math.uc.edu

DOI:
http://dx.doi.org/10.1090/S0002-9939-2011-11288-7

Keywords:
Elliptic system at resonance,
existence of solutions

Received by editor(s):
February 25, 2011

Published electronically:
November 21, 2011

Communicated by:
Walter Craig

Article copyright:
© Copyright 2011
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.