Some regularity results for  a singular elliptic problem

Brahim  Bougherara; Jacques  Giacomoni; Jesus  Hernández

doi:10.3934/proc.2015.0142

Article Contents

2015, Volume 2015: 142-150. Doi: 10.3934/proc.2015.0142 open access

This issue Previous Article Analysis of the archetypal functional equation in the non-critical case Next Article Classical and nonclassical symmetries and exact solutions for a generalized Benjamin equation

Some regularity results for a singular elliptic problem

1.
LMAP-UMR 5142, Bâtiment IPRA, Avenue de l'Université - BP 1155 64013 Pau Cedex
2.
Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid

Received: August 31, 2014

Revised: January 31, 2015

Published: October 2015

Abstract Related Papers Cited by

Abstract

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:

Mathematics Subject Classification: Primary: 35J35, 35J50; Secondary: 35R05.

Citation:

Related Papers

Cited by

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.