Abstract
We prove existence, regularity and nonexistence results for problems whose model is
with zero Dirichlet conditions on the boundary of an open, bounded subset Ω of \({\mathbb{R}^{N}}\). Here γ > 0 and f is a nonnegative function on Ω. Our results will depend on the summability of f in some Lebesgue spaces, and on the values of γ (which can be equal, larger or smaller than 1).
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Alves C.O., Goncalves J.V., Maia L.: Singular nonlinear elliptic equations in \({\mathbb{R}^N}\). Abstr. Appl. Anal. 3, 411–423 (1998)
Arcoya D., Carmona J., Leonori T., Martínez-Aparicio P., Orsina L., Petitta F.: Existence and nonexistence of solutions for singular quadratic quasilinear equations. J. Differential Equations 246, 4006–4042 (2009)
Boccardo L.: Dirichlet problems with singular and gradient quadratic lower order terms. ESAIM Control Optim. Calc. Var. 14, 411–426 (2008)
Boccardo L., Gallouët T.: Nonlinear elliptic equations with right hand side measures. Comm. Partial Differential Equations 17, 641–655 (1992)
Boccardo L., Gallouët T., Orsina L.: Existence and uniqueness of entropy solutions for nonlinear elliptic equations with measure data. Ann. Inst. H. Poincaré Anal. Non Linéaire 13, 539–551 (1996)
Boccardo L., Orsina L.: Sublinear equations in L s. Houston J. Math. 20, 99–114 (1994)
Canino A., Degiovanni M.: A variational approach to a class of singular semilinear elliptic equations. J. Convex Anal. 11, 147–162 (2004)
Coclite M.M., Palmieri G.: On a singular nonlinear Dirichlet problem. Comm. Partial Differential Equations 14, 1315–1327 (1989)
Crandall M.G., Rabinowitz P.H., Tartar L.: On a Dirichlet problem with a singular nonlinearity. Comm. Partial Differential Equations 2, 193–222 (1977)
Dal Maso G., Murat F., Orsina L., Prignet A.: Renormalized solutions of elliptic equations with general measure data. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 28, 741–808 (1999)
Hirano N., Saccon C., Shioji N.: Existence of multiple positive solutions for singular elliptic problems with concave and convex nonlinearities. Adv. Differential Equations 9, 197–220 (2004)
Lair A.V., Shaker A.W.: Entire solution of a singular semilinear elliptic problem. J. Math. Anal. Appl. 200, 498–505 (1996)
Lair A.V., Shaker A.W.: Classical and weak solutions of a singular semilinear elliptic problem. J. Math. Anal. Appl. 211, 371–385 (1997)
Lazer A.C., McKenna P.J.: On a singular nonlinear elliptic boundary-value problem. Proc. Amer. Math. Soc. 111, 721–730 (1991)
Martínez-Aparicio, P.: Singular Dirichlet problems with quadratic gradient (preprint)
Stampacchia G.: Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus. Ann. Inst. Fourier (Grenoble) 15, 189–258 (1965)
Stuart C.A.: Existence and approximation of solutions of non-linear elliptic equations. Math. Z. 147, 53–63 (1976)
Zhang Z., Cheng J.: Existence and optimal estimates of solutions for singular nonlinear Dirichlet problems. Nonlinear Anal. 57, 473–484 (2004)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Boccardo, L., Orsina, L. Semilinear elliptic equations with singular nonlinearities. Calc. Var. 37, 363–380 (2010). https://doi.org/10.1007/s00526-009-0266-x
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00526-009-0266-x