A Singular nonlinear problems with natural growth in the gradient
Abstract
In this paper, we consider the equation 
with boundary conditions 
where 
is an open bounded subset of 
is a Leray-Lions operator defined on is a characteristic function,
and 
is a Carathéodory function such that
sign
For
and 
sufficiently small, we prove the existence of at least one solution u of this problem which is moreover such that the function
belongs to
for some
This solution satisfies some a priori estimates in
Keyword : nonlinear problems, existence, singularity
How to Cite
Hamour, B. (2024). A Singular nonlinear problems with natural growth in the gradient. Mathematical Modelling and Analysis, 29(2), 367–386. https://doi.org/10.3846/mma.2024.17948
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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D. Arcoya, L. Boccardo, T. Leonori and A. Porretta. Some elliptic problems with singular natural growth lower order terms. J. Diff. Equ., 249(11):2771– 2795, 2010. https://doi.org/10.1016/j.jde.2010.05.009
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A. Bensoussan, L. Boccardo and F. Murat. On a nonlinear partial differential equation having natural growth terms and unbounded solution. Ann. Inst. H. Poincaré Analyse non linéaire, 5(4):347–364, 1988. https://doi.org/10.1016/s0294-1449(16)30342-0
L. Boccardo. Dirichlet problems with singular and gradient quadratic lower order terms. ESSAIM: Control Optim. Calc. Var., 14(3):411–426, 2008. https://doi.org/10.1051/cocv:2008031
L. Boccardo and G. Croce. The impact of a lower order term in a Dirichlet problem with a singular nonlinarity. Portugaliæ Mathematica., 76(3/4):407–415, 2019. https://doi.org/10.4171/PM/2041
L. Boccardo and L. Orsina. Semilinear elliptic equations singular nonlinearities. Cal. Var., 37:363–380, 2010. https://doi.org/10.1007/s00526-009-0266-x
J. Carmona, A.J. Martinez Aparicio, P.J. Martinez Aparicio and M. MartinezTeruel. Regularizing effect in singular semilinear problems. Math. Mod. Anal., 28(4):561–580, 2023. https://doi.org/10.3846/mma.2023.18616
M.G. Crandal, P.H. Rabinowitz and L. Tartar. On a Dirichlet problem with a sigular nonlinearity. Comm. Part. Diff. Equ., 2(2):193–222, 1977. https://doi.org/10.1080/03605307708820029
R. Durastanti. Asymptotic behavior and existence of solutions for singular elliptic equations. Ann. Mat. Pura Appl., 199:925–954, 2001. https://doi.org/10.1007/s10231-019-00906-0
V. Ferone and F. Murat. Quasilinear problems having quadratic growth in the gradient: an existence result when the source term is small. Équations aux dérivées partielles et applications, 497–515, 1998.
V. Ferone and F. Murat. Nonlinear problems having natural growth in the gradient: an existence result when the source terms are small. Nonlinear Anal., 42(7):1309–1326, 2000. https://doi.org/10.1016/S0362-546X(99)00165-0
D. Giachetti, P.J. Martinez-Aparicio and F. Murat. A semilinear elliptic equation with a mild singularity at u = 0: existence and homgenization. J. Maths. Pures Appl., 107(1):41–77, 2017. https://doi.org/10.1016/j.matpur.2016.04.007
D. Giachetti, P.J. Martinez-Aparicio and F. Murat. Definition, existence, stability and uniqueness of the solution to semilinear elliptic problem with a strong singularity at u = 0. Ann. Sc. Norm. Super. Pisa Cl.Sci., 18(4):1395–1442, 2018. https://doi.org/10.2422/2036-2145.201612 008
D. Giachetti, F. Petitta and S. Segura de León. A priori estimates for elliptic problems with a strongly singular gradient term and a general datum. Diff. Int. Equ., 26(9-10):913–948, 2013. https://doi.org/10.57262/die/1372858556
D. Giachetti, F. Petitta and S. Segura De León. Elliptic equations having a singular quadratic gradient term and a changing sign datum. Comm. on Pure and Appl. Anal., 11(5):1875–1895, 2012. https://doi.org/10.3934/cpaa.2012.11.1875
B. Hamour. Singular quasilinear problems with quadratic growth in the gradient. Le Matematiche, 77(2):265–292, 2022.
B. Hamour and F. Murat. Quasilinear problems involving a perturbation with quadratic growth in the gradient and a noncoercive zeroth order term. Rend. Lincei Mat. Appl., 27(2):195–233, 2016. https://doi.org/10.4171/RLM/731
J. Leray and J.-L. Lions. Quelques résultats de Višik sur les problèmes elliptiques non linéaires par les méthodes de Minty-Browder. Bull. Soc. Math. France 93, pp. 97–107, 1965. https://doi.org/10.24033/bsmf.1617
J.-L. Lions. Quelques méthodes de réesolution des problèmes aux limites non linéaires. Dunod, 1969.
A. Marah, H. Redwane and K. Zaki. Nonlinear elliptic equations with unbounded coefficient and singular nonlinear elliptic equations with unbounded coefficient and singular. J. Fixed Point Theory Appl., 22(3(68)):18 pp, 2020. https://doi.org/10.1007/s11784-020-00804-6
F. Oliva and F. Petitta. Finite and infinite energy solutions of singular elliptic problems: existence and uniqueness. J. Diff. Equ., 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, 15(1):189–257, 1965. https://doi.org/10.5802/aif.204
Z. Zhang. Two classes of nonlinear singular Dirichlet problems with natural growth: existence and asymptotic behavior. Adv. Nonlinear Stud., 20(1):77–93, 2020. https://doi.org/10.1515/ans-2019-2054
B. Abdellaoui, D. Giachetti, I. Peral and M. Walias. Elliptic problems with nonlinear terms depending on the gradient and singular on the boundary. Nonlinear. Anal., 74(4):1355–1371, 2011. https://doi.org/10.1016/j.na.2010.10.008
D. Arcoya, L. Boccardo, T. Leonori and A. Porretta. Some elliptic problems with singular natural growth lower order terms. J. Diff. Equ., 249(11):2771– 2795, 2010. https://doi.org/10.1016/j.jde.2010.05.009
D. Arcoya, J. Carmona, T. Leonori, P.J. Martinez-Aparicio, L. Orsina and F. Petitta. Existence and nonexistence of solutions for singular quadratic quasilinear equations. J. Diff. Equ., 246(10):4006–4042, 2009. https://doi.org/10.1016/j.jde.2009.01.016
A. Bensoussan, L. Boccardo and F. Murat. On a nonlinear partial differential equation having natural growth terms and unbounded solution. Ann. Inst. H. Poincaré Analyse non linéaire, 5(4):347–364, 1988. https://doi.org/10.1016/s0294-1449(16)30342-0
L. Boccardo. Dirichlet problems with singular and gradient quadratic lower order terms. ESSAIM: Control Optim. Calc. Var., 14(3):411–426, 2008. https://doi.org/10.1051/cocv:2008031
L. Boccardo and G. Croce. The impact of a lower order term in a Dirichlet problem with a singular nonlinarity. Portugaliæ Mathematica., 76(3/4):407–415, 2019. https://doi.org/10.4171/PM/2041
L. Boccardo and L. Orsina. Semilinear elliptic equations singular nonlinearities. Cal. Var., 37:363–380, 2010. https://doi.org/10.1007/s00526-009-0266-x
J. Carmona, A.J. Martinez Aparicio, P.J. Martinez Aparicio and M. MartinezTeruel. Regularizing effect in singular semilinear problems. Math. Mod. Anal., 28(4):561–580, 2023. https://doi.org/10.3846/mma.2023.18616
M.G. Crandal, P.H. Rabinowitz and L. Tartar. On a Dirichlet problem with a sigular nonlinearity. Comm. Part. Diff. Equ., 2(2):193–222, 1977. https://doi.org/10.1080/03605307708820029
R. Durastanti. Asymptotic behavior and existence of solutions for singular elliptic equations. Ann. Mat. Pura Appl., 199:925–954, 2001. https://doi.org/10.1007/s10231-019-00906-0
V. Ferone and F. Murat. Quasilinear problems having quadratic growth in the gradient: an existence result when the source term is small. Équations aux dérivées partielles et applications, 497–515, 1998.
V. Ferone and F. Murat. Nonlinear problems having natural growth in the gradient: an existence result when the source terms are small. Nonlinear Anal., 42(7):1309–1326, 2000. https://doi.org/10.1016/S0362-546X(99)00165-0
D. Giachetti, P.J. Martinez-Aparicio and F. Murat. A semilinear elliptic equation with a mild singularity at u = 0: existence and homgenization. J. Maths. Pures Appl., 107(1):41–77, 2017. https://doi.org/10.1016/j.matpur.2016.04.007
D. Giachetti, P.J. Martinez-Aparicio and F. Murat. Definition, existence, stability and uniqueness of the solution to semilinear elliptic problem with a strong singularity at u = 0. Ann. Sc. Norm. Super. Pisa Cl.Sci., 18(4):1395–1442, 2018. https://doi.org/10.2422/2036-2145.201612 008
D. Giachetti, F. Petitta and S. Segura de León. A priori estimates for elliptic problems with a strongly singular gradient term and a general datum. Diff. Int. Equ., 26(9-10):913–948, 2013. https://doi.org/10.57262/die/1372858556
D. Giachetti, F. Petitta and S. Segura De León. Elliptic equations having a singular quadratic gradient term and a changing sign datum. Comm. on Pure and Appl. Anal., 11(5):1875–1895, 2012. https://doi.org/10.3934/cpaa.2012.11.1875
B. Hamour. Singular quasilinear problems with quadratic growth in the gradient. Le Matematiche, 77(2):265–292, 2022.
B. Hamour and F. Murat. Quasilinear problems involving a perturbation with quadratic growth in the gradient and a noncoercive zeroth order term. Rend. Lincei Mat. Appl., 27(2):195–233, 2016. https://doi.org/10.4171/RLM/731
J. Leray and J.-L. Lions. Quelques résultats de Višik sur les problèmes elliptiques non linéaires par les méthodes de Minty-Browder. Bull. Soc. Math. France 93, pp. 97–107, 1965. https://doi.org/10.24033/bsmf.1617
J.-L. Lions. Quelques méthodes de réesolution des problèmes aux limites non linéaires. Dunod, 1969.
A. Marah, H. Redwane and K. Zaki. Nonlinear elliptic equations with unbounded coefficient and singular nonlinear elliptic equations with unbounded coefficient and singular. J. Fixed Point Theory Appl., 22(3(68)):18 pp, 2020. https://doi.org/10.1007/s11784-020-00804-6
F. Oliva and F. Petitta. Finite and infinite energy solutions of singular elliptic problems: existence and uniqueness. J. Diff. Equ., 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, 15(1):189–257, 1965. https://doi.org/10.5802/aif.204
Z. Zhang. Two classes of nonlinear singular Dirichlet problems with natural growth: existence and asymptotic behavior. Adv. Nonlinear Stud., 20(1):77–93, 2020. https://doi.org/10.1515/ans-2019-2054