# American Institute of Mathematical Sciences

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

## Some regularity results for a singular elliptic problem

 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).
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., (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.   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.   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.   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.   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.   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 (2012), 7.   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.   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 (2012), 7.   Google Scholar [10] S. M. Gomes, On a singular nonlinear elliptic problem,, SIAM J. Math. Anal., 17 (1986), 1359.   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.   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.   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.   Google Scholar [14] G. Lieberman, Boundary regularity for solutions of degenerate elliptic equations., Nonlinear Analysis, 12 (1988), 1203.   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.   Google Scholar [16] J. Serrin, Local behavior of solutions of quasi-linear equations., Acta Math., 111 (1964), 247.   Google Scholar [17] P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations,, J. Differential Equations, 51 (1984), 126.   Google Scholar [18] J.L. Vázquez, A strong maximum principle for some quasilinear equations,, Appl. Math. Opt., 12 (1984), 1992.   Google Scholar [19] Z. Zhang and J. Cheng, Existence and optimal estimates of solutions for singular nonlinear Dirichlet problems,, Nonlinear Anal., 57 (2004), 473.   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., (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.   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.   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.   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.   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.   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 (2012), 7.   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.   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 (2012), 7.   Google Scholar [10] S. M. Gomes, On a singular nonlinear elliptic problem,, SIAM J. Math. Anal., 17 (1986), 1359.   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.   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.   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.   Google Scholar [14] G. Lieberman, Boundary regularity for solutions of degenerate elliptic equations., Nonlinear Analysis, 12 (1988), 1203.   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.   Google Scholar [16] J. Serrin, Local behavior of solutions of quasi-linear equations., Acta Math., 111 (1964), 247.   Google Scholar [17] P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations,, J. Differential Equations, 51 (1984), 126.   Google Scholar [18] J.L. Vázquez, A strong maximum principle for some quasilinear equations,, Appl. Math. Opt., 12 (1984), 1992.   Google Scholar [19] Z. Zhang and J. Cheng, Existence and optimal estimates of solutions for singular nonlinear Dirichlet problems,, Nonlinear Anal., 57 (2004), 473.   Google Scholar
 [1] Yongxiu Shi, Haitao Wan. Refined asymptotic behavior and uniqueness of large solutions to a quasilinear elliptic equation in a borderline case. Electronic Research Archive, , () : -. doi: 10.3934/era.2020119 [2] Neil S. Trudinger, Xu-Jia Wang. Quasilinear elliptic equations with signed measure. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 477-494. doi: 10.3934/dcds.2009.23.477 [3] Nahed Naceur, Nour Eddine Alaa, Moez Khenissi, Jean R. Roche. Theoretical and numerical analysis of a class of quasilinear elliptic equations. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 723-743. doi: 10.3934/dcdss.2020354 [4] Christian Clason, Vu Huu Nhu, Arnd Rösch. Optimal control of a non-smooth quasilinear elliptic equation. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020052 [5] Tuoc Phan, Grozdena Todorova, Borislav Yordanov. Existence uniqueness and regularity theory for elliptic equations with complex-valued potentials. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1071-1099. doi: 10.3934/dcds.2020310 [6] Stefano Bianchini, Paolo Bonicatto. Forward untangling and applications to the uniqueness problem for the continuity equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020384 [7] Xinfu Chen, Huiqiang Jiang, Guoqing Liu. Boundary spike of the singular limit of an energy minimizing problem. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3253-3290. doi: 10.3934/dcds.2020124 [8] Maho Endo, Yuki Kaneko, Yoshio Yamada. Free boundary problem for a reaction-diffusion equation with positive bistable nonlinearity. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3375-3394. doi: 10.3934/dcds.2020033 [9] Jianli Xiang, Guozheng Yan. The uniqueness of the inverse elastic wave scattering problem based on the mixed reciprocity relation. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021004 [10] Yichen Zhang, Meiqiang Feng. A coupled $p$-Laplacian elliptic system: Existence, uniqueness and asymptotic behavior. Electronic Research Archive, 2020, 28 (4) : 1419-1438. doi: 10.3934/era.2020075 [11] Shenglan Xie, Maoan Han, Peng Zhu. A posteriori error estimate of weak Galerkin fem for second order elliptic problem with mixed boundary condition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020340 [12] Yunfeng Jia, Yi Li, Jianhua Wu, Hong-Kun Xu. Cauchy problem of semilinear inhomogeneous elliptic equations of Matukuma-type with multiple growth terms. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3485-3507. doi: 10.3934/dcds.2019227 [13] Eduard Marušić-Paloka, Igor Pažanin. Homogenization and singular perturbation in porous media. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020279 [14] Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020345 [15] Craig Cowan, Abdolrahman Razani. Singular solutions of a Lane-Emden system. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 621-656. doi: 10.3934/dcds.2020291 [16] Jiaquan Liu, Xiangqing Liu, Zhi-Qiang Wang. Sign-changing solutions for a parameter-dependent quasilinear equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020454 [17] Roland Schnaubelt, Martin Spitz. Local wellposedness of quasilinear Maxwell equations with absorbing boundary conditions. Evolution Equations & Control Theory, 2021, 10 (1) : 155-198. doi: 10.3934/eect.2020061 [18] Zhouxin Li, Yimin Zhang. Ground states for a class of quasilinear Schrödinger equations with vanishing potentials. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020298 [19] Boris Andreianov, Mohamed Maliki. On classes of well-posedness for quasilinear diffusion equations in the whole space. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 505-531. doi: 10.3934/dcdss.2020361 [20] Shao-Xia Qiao, Li-Jun Du. Propagation dynamics of nonlocal dispersal equations with inhomogeneous bistable nonlinearity. Electronic Research Archive, , () : -. doi: 10.3934/era.2020116

Impact Factor: