Abstract
We consider a semilinear elliptic problem, with principal part possibly non symmetric, having a singular nonlinear term which is convex near zero and concave at infinity. We prove the existence of two positive solutions when a suitable parameter is small and a nonexistence result when the parameter is large. These results are closely related to a well known paper by Ambrosetti, Brezis, Cerami, where the principal part is the Laplace operator and the non linearity has no singularity. We use monotonicity arguments to get rid of the singular term. Since the problem has no variational structure, we use degree arguments to exploit the topological features of the problem; in particular, to use continuation arguments, we prove a global bound for all positive solutions.
Mathematics Subject Classification (2010). 35J75, 35J87, 47H05, 47H11.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
A. Ambrosetti, H. Brezis, and G. Cerami. Combined effects of concave and convex nonlinearities in some elliptic problems. J. Funct. Anal., 122:519–543, 1994.
L. Boccardo. A Dirichlet problem with singular and supercritical nonlinearities. Nonlinear Anal. Th. Meth. Appl., 75:4436–4440, 2012.
H. Brezis. Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. 1973.
M.G. Crandall, P.H. Rabinowitz, and L. Tartar. On a Dirichlet problem with a singular nonlinearity. Comm. Part. Diff. Eq., 2:193–222, 1977.
B. Gidas and J. Spruck. A priori bounds for positive solutions of nonlinear elliptic equations. Comm. Partial Differ. Equations, 6 (8):883–901, 1981.
D. Gilbarg and N.S. Trudinger. Elliptic Partial Differential Equations of Second Order. 1998.
J. Hernandez, F. Mancebo, and J. Vega. Positive solutions for singular nonlinear elliptic equations. Proc. Royal Soc. Edinburgh, 137A:41–62, 2007.
N. Hirano, C. Saccon, and N. Shioji. Existence of multiple positive solutions for singular elliptic problems with concave and convex nonlinearities. Adv. Diff. Eq., 9(1-2):197–220, 2004.
N. Hirano, C. Saccon, and N. Shioji. Brezis–Nirenberg type problems and multiplicity of positive solutions for a singular elliptic problem. J. Diff. Eq., 245:1997–2037, 2008.
A.I. Nazarov. A centennial of the Zaremba–Hopf–Oleinik lemma. SIAM J. Math. Anal., 44:437–453, 2012.
C. Saccon. A variational approach to a class of singular semilinear elliptic equations. EJDE, to appear, 2013. Proceedings of the Conference “Variational and Topological Methods”, Flagstaff, June 6–9, 2012.
G. Stampacchia. “Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus”. Ann. Inst. Fourier, 15(1):189–258, 1965.
C.A. Stuart. Existence and approximation of solutions of non-linear elliptic equations. Math. Z., 137A:53–63, 1976.
N.S. Trudinger. Linear elliptic operators with measurable coefficients. Ann. Scuola Norm. Sup. Pisa, 27(2):265–308, 1973.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Saccon, C. (2014). Multiple Positive Solutions for a Nonsymmetric Elliptic Problem with Concave Convex Nonlinearity. In: de Figueiredo, D., do Ó, J., Tomei, C. (eds) Analysis and Topology in Nonlinear Differential Equations. Progress in Nonlinear Differential Equations and Their Applications, vol 85. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-04214-5_23
Download citation
DOI: https://doi.org/10.1007/978-3-319-04214-5_23
Published:
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-319-04213-8
Online ISBN: 978-3-319-04214-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)