American Institute of Mathematical Sciences

Advanced
2015, 2015(special): 142-150. doi: 10.3934/proc.2015.0142

Some regularity results for a singular elliptic problem

Brahim Bougherara 1, , Jacques Giacomoni 1, and  Jesus Hernández 2,

1. 

LMAP-UMR 5142, Bâtiment IPRA, Avenue de l'Université - BP 1155 64013 Pau Cedex, France, France
2. 

Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain

Received  September 2014 Revised  February 2015 Published  November 2015

In the present paper we investigate the following singular elliptic problem with $p$-Laplacian operator: \begin{equation*} (P)\qquad \left \{ \begin{array}{l} -\Delta_p u = \frac{K(x)}{ u^{\alpha}}\quad \text{ in } \Omega \\ u = 0\ \text{ on } \partial\Omega,\ u>0 \text{ on } \Omega, \end{array} \right . \end{equation*} where $\Omega$ is a regular bounded domain of $\mathbb R^{N}$, $\alpha\in\mathbb R$, $K\in L^\infty_{\rm loc}(\Omega)$ a non-negative function. We discuss below the existence, the regularity and the uniqueness of a weak solution $u$ to the problem (P).
Keywords: regulariy., singular nonlinearity, uniqueness, Quasilinear elliptic problem.
Mathematics Subject Classification: Primary: 35J35, 35J50; Secondary: 35R0.
Citation: Brahim Bougherara, Jacques Giacomoni, Jesus Hernández. Some regularity results for a singular elliptic problem. Conference Publications, 2015, 2015 (special) : 142-150. doi: 10.3934/proc.2015.0142
References:
[1]

L. Bougherara, J. Giacomoni and J. Hernández, Existence and regularity of weak solutions for singular semilinear elliptic problems, Preprint 2014.

[2]

M. G. Crandall, P. H. Rabinowitz, and L. Tartar, On a Dirichlet problem with a singular nonlinearity, Comm. Partial Differential Equations, 2 (1977), 193-222.

[3]

M. Del Pino, A global estimate for the gradient in a singular elliptic boundary value problem, Proc. Roy. Soc. Edinburgh Sect. A, 122 (1992), 341-352.

[4]

J. I. Díaz, J. Hernández and J. M. Rakotoson, On very weak positive solutions to some semilinear elliptic problems with simultaneous singular nonlinear and spatial dependence terms, Milan J. Math., 79 (2011), 233-245.

[5]

J. I. Díaz and J. M. Rakotoson, On the differentiability of very weak solutions with right-hand side data integrable with respect to the distance to the boundary, J. Funct. Anal., 257 (2009), 807-831.

[6]

J. I. Díaz and J. E. Saá, Existence et unicité de solutions positives pour certaines équations elliptiques quasilinéaires. C. R. Acad. Sci. Paris Sér. I Math., 305 (1988), 321-324.

[7]

J. Giacomoni, H. Maagli and P. Sauvy, Existence of compact support solutions for a quasilinear and singular problem, Differential Integral Equations, 25(7-8) (2012), 629-656.

[8]

J. Giacomoni, I. Schindler and P. Takáč, Sobolev versus Hölder local minimizers and existence of multiple solutions for a singular quasilinear equation, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 6 (2007), 117-158.

[9]

J. Giacomoni, I. Schindler and P. Takáč, $C^{0,\beta}$-regularity and singular quasilinear elliptic equations, C. R. Math. Acad. Sci. Paris, 350(7-8) (2012), 383-388.

[10]

S. M. Gomes, On a singular nonlinear elliptic problem, SIAM J. Math. Anal., 17 (1986), 1359-1369.

[11]

C. Gui and F. H. Lin, Regularity of an elliptic problem with a singular nonlinearity, Proc. Roy. Soc. Edinburgh Sect. A, 123 (1993), 1021-1029.

[12]

J. Hernández, F. Mancebo and J. M. Vega, Positive solutions for singular nonlinear elliptic equations, Proc. Roy. Soc. Edinburgh Sect. A, 137 (2007), 41-62.

[13]

A. C. Lazer and P. J. McKenna, On a singular nonlinear elliptic boundary-value problem, Proc. Amer. Math. Soc., 111 (1991), 721-730.

[14]

G. Lieberman, Boundary regularity for solutions of degenerate elliptic equations. Nonlinear Analysis, 12 (1988), 1203-1219.

[15]

P. Lindqvist, On the equation div $(|\nabla u|^{p-2}\nabla u)+\lambda|u|^{p-2}u=0$, Proc. Amer. Math. Soc., 109 (1990), 157-164.

[16]

J. Serrin, Local behavior of solutions of quasi-linear equations. Acta Math., 111 (1964), 247-302.

[17]

P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations, 51 (1984), 126-150.

[18]

J.L. Vázquez, A strong maximum principle for some quasilinear equations, Appl. Math. Opt., 12 (1984), 1992-2002.

[19]

Z. Zhang and J. Cheng, Existence and optimal estimates of solutions for singular nonlinear Dirichlet problems, Nonlinear Anal., 57 (2004), 473-484.

show all references

References:
[1]

L. Bougherara, J. Giacomoni and J. Hernández, Existence and regularity of weak solutions for singular semilinear elliptic problems, Preprint 2014.

[2]

M. G. Crandall, P. H. Rabinowitz, and L. Tartar, On a Dirichlet problem with a singular nonlinearity, Comm. Partial Differential Equations, 2 (1977), 193-222.

[3]

M. Del Pino, A global estimate for the gradient in a singular elliptic boundary value problem, Proc. Roy. Soc. Edinburgh Sect. A, 122 (1992), 341-352.

[4]

J. I. Díaz, J. Hernández and J. M. Rakotoson, On very weak positive solutions to some semilinear elliptic problems with simultaneous singular nonlinear and spatial dependence terms, Milan J. Math., 79 (2011), 233-245.

[5]

J. I. Díaz and J. M. Rakotoson, On the differentiability of very weak solutions with right-hand side data integrable with respect to the distance to the boundary, J. Funct. Anal., 257 (2009), 807-831.

[6]

J. I. Díaz and J. E. Saá, Existence et unicité de solutions positives pour certaines équations elliptiques quasilinéaires. C. R. Acad. Sci. Paris Sér. I Math., 305 (1988), 321-324.

[7]

J. Giacomoni, H. Maagli and P. Sauvy, Existence of compact support solutions for a quasilinear and singular problem, Differential Integral Equations, 25(7-8) (2012), 629-656.

[8]

J. Giacomoni, I. Schindler and P. Takáč, Sobolev versus Hölder local minimizers and existence of multiple solutions for a singular quasilinear equation, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 6 (2007), 117-158.

[9]

J. Giacomoni, I. Schindler and P. Takáč, $C^{0,\beta}$-regularity and singular quasilinear elliptic equations, C. R. Math. Acad. Sci. Paris, 350(7-8) (2012), 383-388.

[10]

S. M. Gomes, On a singular nonlinear elliptic problem, SIAM J. Math. Anal., 17 (1986), 1359-1369.

[11]

C. Gui and F. H. Lin, Regularity of an elliptic problem with a singular nonlinearity, Proc. Roy. Soc. Edinburgh Sect. A, 123 (1993), 1021-1029.

[12]

J. Hernández, F. Mancebo and J. M. Vega, Positive solutions for singular nonlinear elliptic equations, Proc. Roy. Soc. Edinburgh Sect. A, 137 (2007), 41-62.

[13]

A. C. Lazer and P. J. McKenna, On a singular nonlinear elliptic boundary-value problem, Proc. Amer. Math. Soc., 111 (1991), 721-730.

[14]

G. Lieberman, Boundary regularity for solutions of degenerate elliptic equations. Nonlinear Analysis, 12 (1988), 1203-1219.

[15]

P. Lindqvist, On the equation div $(|\nabla u|^{p-2}\nabla u)+\lambda|u|^{p-2}u=0$, Proc. Amer. Math. Soc., 109 (1990), 157-164.

[16]

J. Serrin, Local behavior of solutions of quasi-linear equations. Acta Math., 111 (1964), 247-302.

[17]

P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations, 51 (1984), 126-150.

[18]

J.L. Vázquez, A strong maximum principle for some quasilinear equations, Appl. Math. Opt., 12 (1984), 1992-2002.

[19]

Z. Zhang and J. Cheng, Existence and optimal estimates of solutions for singular nonlinear Dirichlet problems, Nonlinear Anal., 57 (2004), 473-484.

[1]

Boumediene Abdellaoui, Ahmed Attar. Quasilinear elliptic problem with Hardy potential and singular term. Communications on Pure and Applied Analysis, 2013, 12 (3) : 1363-1380. doi: 10.3934/cpaa.2013.12.1363
[2]

Zongming Guo, Xuefei Bai. On the global branch of positive radial solutions of an elliptic problem with singular nonlinearity. Communications on Pure and Applied Analysis, 2008, 7 (5) : 1091-1107. doi: 10.3934/cpaa.2008.7.1091
[3]

Baishun Lai, Qing Luo. Regularity of the extremal solution for a fourth-order elliptic problem with singular nonlinearity. Discrete and Continuous Dynamical Systems, 2011, 30 (1) : 227-241. doi: 10.3934/dcds.2011.30.227
[4]

Zongming Guo, Juncheng Wei. Asymptotic behavior of touch-down solutions and global bifurcations for an elliptic problem with a singular nonlinearity. Communications on Pure and Applied Analysis, 2008, 7 (4) : 765-786. doi: 10.3934/cpaa.2008.7.765
[5]

Zongming Guo, Long Wei. A fourth order elliptic equation with a singular nonlinearity. Communications on Pure and Applied Analysis, 2014, 13 (6) : 2493-2508. doi: 10.3934/cpaa.2014.13.2493
[6]

Prashanta Garain, Tuhina Mukherjee. Quasilinear nonlocal elliptic problems with variable singular exponent. Communications on Pure and Applied Analysis, 2020, 19 (11) : 5059-5075. doi: 10.3934/cpaa.2020226
[7]

Marino Badiale, Michela Guida, Sergio Rolando. Radial quasilinear elliptic problems with singular or vanishing potentials. Communications on Pure and Applied Analysis, 2022, 21 (1) : 23-46. doi: 10.3934/cpaa.2021165
[8]

Uriel Kaufmann, Humberto Ramos Quoirin, Kenichiro Umezu. Uniqueness and sign properties of minimizers in a quasilinear indefinite problem. Communications on Pure and Applied Analysis, , () : -. doi: 10.3934/10.3934/cpaa.2021078
[9]

Uriel Kaufmann, Humberto Ramos Quoirin, Kenichiro Umezu. Uniqueness and sign properties of minimizers in a quasilinear indefinite problem. Communications on Pure and Applied Analysis, 2021, 20 (6) : 2313-2322. doi: 10.3934/cpaa.2021078
[10]

Fouad Hadj Selem, Hiroaki Kikuchi, Juncheng Wei. Existence and uniqueness of singular solution to stationary Schrödinger equation with supercritical nonlinearity. Discrete and Continuous Dynamical Systems, 2013, 33 (10) : 4613-4626. doi: 10.3934/dcds.2013.33.4613
[11]

Giuseppe Maria Coclite, Mario Michele Coclite. On a Dirichlet problem in bounded domains with singular nonlinearity. Discrete and Continuous Dynamical Systems, 2013, 33 (11&12) : 4923-4944. doi: 10.3934/dcds.2013.33.4923
[12]

Kazuaki Taira. The hypoelliptic Robin problem for quasilinear elliptic equations. Discrete and Continuous Dynamical Systems - S, 2020, 13 (5) : 1601-1618. doi: 10.3934/dcdss.2020091
[13]

Galina V. Grishina. On positive solution to a second order elliptic equation with a singular nonlinearity. Communications on Pure and Applied Analysis, 2010, 9 (5) : 1335-1343. doi: 10.3934/cpaa.2010.9.1335
[14]

Zongming Guo, Zhongyuan Liu, Juncheng Wei, Feng Zhou. Bifurcations of some elliptic problems with a singular nonlinearity via Morse index. Communications on Pure and Applied Analysis, 2011, 10 (2) : 507-525. doi: 10.3934/cpaa.2011.10.507
[15]

Zongming Guo, Yunting Yu. Boundary value problems for a semilinear elliptic equation with singular nonlinearity. Communications on Pure and Applied Analysis, 2016, 15 (2) : 399-412. doi: 10.3934/cpaa.2016.15.399
[16]

Miao Chen, Youyan Wan, Chang-Lin Xiang. Local uniqueness problem for a nonlinear elliptic equation. Communications on Pure and Applied Analysis, 2020, 19 (2) : 1037-1055. doi: 10.3934/cpaa.2020048
[17]

Minh-Phuong Tran, Thanh-Nhan Nguyen. Pointwise gradient bounds for a class of very singular quasilinear elliptic equations. Discrete and Continuous Dynamical Systems, 2021, 41 (9) : 4461-4476. doi: 10.3934/dcds.2021043
[18]

Luiz F. O. Faria. Existence and uniqueness of positive solutions for singular biharmonic elliptic systems. Conference Publications, 2015, 2015 (special) : 400-408. doi: 10.3934/proc.2015.0400
[19]

Yen-Lin Wu, Zhi-You Chen, Jann-Long Chern, Y. Kabeya. Existence and uniqueness of singular solutions for elliptic equation on the hyperbolic space. Communications on Pure and Applied Analysis, 2014, 13 (2) : 949-960. doi: 10.3934/cpaa.2014.13.949
[20]

M. Chuaqui, C. Cortázar, M. Elgueta, J. García-Melián. Uniqueness and boundary behavior of large solutions to elliptic problems with singular weights. Communications on Pure and Applied Analysis, 2004, 3 (4) : 653-662. doi: 10.3934/cpaa.2004.3.653

 Impact Factor: 

Tools

Metrics

  • PDF downloads (113)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy