Positive Solutions for a Singular Nonlinear Problem on a Bounded Domain in R 2

Springer Science and Business Media LLC - Tập 18 - Trang 97-118 - 2003
Noureddine Zeddini1
1Département de Mathématiques, Faculté des Sciences de Tunis Campus Universitaire, Tunis, Tunisia

Tóm tắt

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.

Tài liệu tham khảo

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.