Abstract
For a bounded regular Jordan domain Ω in R 2, we introduce and study a new class of functions K(Ω) related on its Green function G. We exploit the properties of this class to prove the existence and the uniqueness of a positive solution for the singular nonlinear elliptic equation Δu+ϕ(x,u)=0, in D′(Ω), with u=0 on ∂Ω and u∈C―(Ω), where ϕ is a nonnegative Borel measurable function in Ω×(0,∞) that belongs to a convex cone which contains, in particular, all functions ϕ(x,t)=q(x)t −γ,γ>0 with nonnegative functions q∈K(Ω). Some estimates on the solution are also given.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bliedtner, J. and Hansen, W.: Potential Theory. An Analytic and Probabilistic Approach to Balayage, Springer-Verlag, 1986.
Chung, K.L. and Zhao, Z.: From Brownian Motion to Schrödinger's Equation, Springer-Verlag, 1995.
Dautray, R., Lions, J.L. et al.:Analyse mathématique et calcul numérique pour les sciences et les techniques, Coll. CEA, Vol. 2, l'opérateur de Laplace, Masson, 1987.
Dynkin, E.B.: 'A probabilistic approach to one class of nonlinear differential equations', Probab. Theory Related Fields 89 (1991), 89-115.
Edelson, A.L.: 'Entire solutions of a singular semilinear elliptic problem', J. Math. Anal. Appl. 139 (1989), 523-532.
Lair, A.V. and Shaker, A.W.: 'Entire solutions of a singular elliptic problem', J. Math. Anal. Appl. 200 (1996), 498-505.
Lair, A.V. and Shaker, A.W.: 'Classical and weak solutions of a singular semilinear problem', J. Math. Anal. Appl. 211 (1997), 371-385.
Lazer, A.C. and Mckenna, P.J.: 'On a singular nonlinear elliptic boundary value problem', Proc. Amer. Math. Soc. 111 (1991), 721-730.
Maâgli, H. and Masmoudi, S.: 'Sur les solutions d'un opérateur différentiel singulier semilinéaire', Potential Anal. 10 (1999), 289-304.
Maâgli, H. and Selmi, M.: 'Inequalities for the Green function of the fractional Laplacian', to appear.
Port, S.C. and Stone, C.J.: Brownian Motion and Classical Potential Theory, Academic Press, 1978.
Selmi, M.: 'Inequalities for Green functions in a Dini—Jordan domain in R2', Potential Anal. 13(2000),81-102.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Zeddini, N. Positive Solutions for a Singular Nonlinear Problem on a Bounded Domain in R 2 . Potential Analysis 18, 97–118 (2003). https://doi.org/10.1023/A:1020559619108
Issue Date:
DOI: https://doi.org/10.1023/A:1020559619108