Abstract
For a homogeneous Dirichlet problem attached to a semilinear elliptic equation we study the existence and uniqueness of non-negative solutions. Our analysis is based on straightforward use of Schauder’s fixed point principle for existence and comparatively, based on the Banach’s contraction principle and on a generalized maximum principle for the uniqueness. We have obtained two independent conditions for uniqueness of solution. Both depend on the geometry of the domain as well as on the parameters of the problem.
Authors
Tiberiu Popoviciu Institute of Numerical Analysis
Al. Tămăşan
Babeş-Bolyai University, Cluj-Napoca, Romania
Keywords
boundary value problem; elliptic; quadratic nonlinearity; non-negative solution; existence; uniqueness; Schauder fixed point; Banach contraction; generalized maximum ;
References
See the expanding block below.
Cite this paper as
C.I. Gheorghiu, Al. Tămăşan, On the existence and uniqueness of positive solutions of some mildly nonlinear elliptic boundary value problems, Rev. Anal. Numér. Théor. Approx. 24 (1995), pp. 125-129.
About this paper
Journal
Rev. Anal. Numér. Théor. Approx.
Publisher Name
Editions de l’Academie Roumaine
Paper on journal website
Print ISSN
1222-9024
Online ISSN
2457-8126
MR
?
ZBL
?
Google Scholar
?
[1] Shampine, L.F., Wing, G.M., Existence and Uniqueness of Solutions of a Class olf Nonlinear Elliptic Boundary Value Problems, J. Math. Mech., 19 (1970), pp. 971-979.
[2] Rus, I.A., Fixed Points Principles, Ed. Dacia, 1979 (in Romanian).
[3] Petrila T., Gheorghiu, C.I., Finite Elements Method and Applications, Ed. Academiei, Bucureşti 1987 (in Romanian).
[4] Protter, H.M., Weinberger, H.F., Maximum Principles in Differential Equations, Prentice Hall, Inc. 1967.
[5] Kantorovici, L.V., Akilov, G.P., Functional analysis (Romanian translation), Nauka ed. 1977.
[6] Berger, M.S., Fraenkel, L.E., On the Asymptotic Solution of a Nonlinear Dirichlet Problem, J. Math. Mech., 19 (1970), pp. 553-585.