Regularizing effect in singular semilinear problems
Abstract
We analyze how different relations in the lower order terms lead to the same regularizing effect on singular problems whose model is 
in Ω, u = 0 on ∂Ω, where Ω is a bounded open set of 
is a nonnegative function in L1(Ω) and g(x,s) is a Carathéodory function. In a framework where no 
solution is expected, we prove its existence (regularizing effect) whenever the datum f interacts conveniently either with the boundary of the domain or with the lower order term.
Keyword : nonlinear elliptic equations, singular problem, regularizing effect
How to Cite
Carmona, J., Martínez Aparicio, A. J., Martínez-Aparicio, P. J., & Martínez-Teruel, M. (2023). Regularizing effect in singular semilinear problems. Mathematical Modelling and Analysis, 28(4), 561–580. https://doi.org/10.3846/mma.2023.18616
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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C.A. Stuart. Existence and approximation of solutions of non-linear elliptic equations. Math. Z., 147(1):53–63, 1976. https://doi.org/10.1007/BF01214274
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D. Arcoya and L. Moreno-Mérida. Multiplicity of solutions for a Dirichlet problem with a strongly singular nonlinearity. Nonlinear Anal., 95:281–291, 2014. https://doi.org/10.1016/j.na.2013.09.002
L. Boccardo, S. Buccheri and C.A. dos Santos. An elliptic system with singular nonlinearities: existence via non variational arguments. J. Math. Anal. Appl., 516(1):Paper No. 126490, 21 pp., 2022. https://doi.org/10.1016/j.jmaa.2022.126490
L. Boccardo and L. Orsina. Semilinear elliptic equations with singular nonlinearities. Calc. Var. Partial Differential Equations, 37(3-4):363–380, 2010. https://doi.org/10.1007/s00526-009-0266-x
J. Carmona, T. Leonori, S. López-Martínez and P.J. Martínez-Aparicio. Quasilinear elliptic problems with singular and homogeneous lower order terms. Nonlinear Anal., 179:105–130, 2019. https://doi.org/10.1016/j.na.2018.08.002
J. Carmona and P.J. Martínez-Aparicio. A singular semilinear elliptic equation with a variable exponent. Adv. Nonlinear Stud., 16(3):491–498, 2016. https://doi.org/10.1515/ans-2015-5039
M.G. Crandall, P.H. Rabinowitz and L. Tartar. On a Dirichlet problem with a singular nonlinearity. Comm. Partial Differential Equations, 2(2):193–222, 1977. https://doi.org/10.1080/03605307708820029
W. Fulks and J.S. Maybee. A singular non-linear equation. Osaka Math. J., 12:1–19, 1960.
D. Giachetti, P.J. Martínez-Aparicio and F. Murat. A semilinear elliptic equation with a mild singularity at u = 0: existence and homogenization. J. Math. Pures Appl. (9), 107(1):41–77, 2017. https://doi.org//10.1016/j.matpur.2016.04.007
D. Giachetti, P.J. Martínez-Aparicio and F. Murat. Definition, existence, stability and uniqueness of the solution to a semilinear elliptic problem with a strong singularity at u = 0. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 18(4):1395–1442, 2018. https://doi.org/10.48550/arXiv.1606.07267
A.V. Lair and A.W. Shaker. Entire solution of a singular semilinear elliptic problem. J. Math. Anal. Appl., 200(2):498–505, 1996. https://doi.org/10.1006/jmaa.1996.0218
A.V. Lair and A.W. Shaker. Classical and weak solutions of a singular semilinear elliptic problem. J. Math. Anal. Appl., 211(2):371–385, 1997. https://doi.org/10.1006/jmaa.1997.5470
A.C. Lazer and P.J. McKenna. On a singular nonlinear elliptic boundary-value problem. Proc. Amer. Math. Soc., 111(3):721–730, 1991. https://doi.org/10.1090/S0002-9939-1991-1037213-9
F. Oliva and F. Petitta. On singular elliptic equations with measure sources. ESAIM Control Optim. Calc. Var., 22(1):289–308, 2016. https://doi.org/10.1051/cocv/2015004
F. Oliva and F. Petitta. Finite and infinite energy solutions of singular elliptic problems: existence and uniqueness. J. Differential Equations, 264(1):311–340, 2018. https://doi.org/10.1016/j.jde.2017.09.008
G. Stampacchia. Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus. Ann. Inst. Fourier (Grenoble), 15(fasc. 1):189–258, 1965.
C.A. Stuart. Existence and approximation of solutions of non-linear elliptic equations. Math. Z., 147(1):53–63, 1976. https://doi.org/10.1007/BF01214274
Z. Zhang and J. Cheng. Existence and optimal estimates of solutions for singular nonlinear Dirichlet problems. Nonlinear Anal., 57(3):473–484, 2004. https://doi.org/10.1016/j.na.2004.02.025