American Institute of Mathematical Sciences

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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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[10]

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

[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. Google Scholar

[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. Google Scholar

[13]

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

[14]

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

[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. Google Scholar

[16]

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

[17]

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

[18]

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

[19]

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

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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[10]

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

[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. Google Scholar

[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. Google Scholar

[13]

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

[14]

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

[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. Google Scholar

[16]

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

[17]

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

[18]

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

[19]

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

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