Abstract
We investigate the homogeneous Dirichlet problem in uniformly convex domains for a large class of degenerate elliptic equations with singular zero order term. In particular we establish sharp existence and uniqueness results of positive viscosity solutions.
Similar content being viewed by others
References
Birindelli, I., Galise, G., Ishii, H.: A family of degenerate elliptic operators: maximum principle and its consequences. Ann. Inst. H. Poincaré Anal. Non Linéaire 35(2), 417–441 (2018)
Birindelli, I., Galise, G., Leoni, F.: Liouville theorems for a family of very degenerate elliptic nonlinear operators. Nonlinear Anal. 161, 198–211 (2017)
Caffarelli, L., Li, Y.Y., Nirenberg, L.: Some remarks on singular solutions of nonlinear elliptic equations, I. J. Fixed Point Theory Appl. 5(2), 353–395 (2009)
Crandall, M.G., Ishii, H., Lions, P.-L.: User’s guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. (N.S.) 27(1), 1–67 (1992)
Crandall, M.G., Rabinowitz, P.H., Tartar, L.: On a Dirichlet problem with a singular nonlinearity. Commun. Partial Differ. Equ. 2(2), 193–222 (1977)
del Pino, M.A.: A global estimate for the gradient in a singular elliptic boundary value problem. Proc. R. Soc. Edinb. Sect. A 122(3–4), 341–352 (1992)
Dolcetta, I.C., Leoni, F., Vitolo, A.: On the inequality \(F(x, D^2u)\ge f(u)+g(u)|Du|^q\). Math. Ann. 365(1–2), 423–448 (2016)
Felmer, P., Quaas, A., Sirakov, B.: Existence and regularity results for fully nonlinear equations with singularities. Math. Ann. 354(1), 377–400 (2012)
Galise, G.: On positive solutions of fully nonlinear degenerate Lane–Emden type equations. J. Differ. Equ. 266(2–3), 1675–1697 (2019)
Harvey, F.R., Lawson Jr., H.B.: Dirichlet duality and the nonlinear Dirichlet problem. Commun. Pure Appl. Math. 62(3), 396–443 (2009)
Harvey, F.R., Lawson Jr., H.B.: \(p\)-convexity, \(p\)-plurisubharmonicity and the Levi problem. Indiana Univ. Math. J. 62(1), 149–169 (2013)
Hernandez, J., Mancebo, F.J.: Singular elliptic and parabolic equations. In: Chipot, M., Quittner, P. (eds.) Handbook of Differential Equations, Stationary Partial Differential Equations, vol. 3, pp. 317–400. Elsevier, Amsterdam (2006)
Lazer, A.C., McKenna, P.J.: On a singular nonlinear elliptic boundary-value problem. Proc. Am. Math. Soc. 111(3), 721–730 (1991)
Stein, E.M.: Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, vol. 30. Princeton University Press, Princeton (1970)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by M. Del Pino.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Birindelli, I., Galise, G. The Dirichlet problem for fully nonlinear degenerate elliptic equations with a singular nonlinearity. Calc. Var. 58, 180 (2019). https://doi.org/10.1007/s00526-019-1629-6
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00526-019-1629-6