A remark on existence and optimal summability of solutions of elliptic problems involving Hardy potential

Previous Article
DCDS Home
This Issue
Next Article
Nodal bubble-tower solutions to radial elliptic problems near criticality

September 2006, 16(3): 513-523. doi: 10.3934/dcds.2006.16.513

A remark on existence and optimal summability of solutions of elliptic problems involving Hardy potential

Lucio Boccardo ^1, , Luigi Orsina ^2, and Ireneo Peral ^3,

1.	Dipartimento di Matematica, Università di Roma 1, Piazza A. Moro 2, 00185 Roma
2.	Dipartimento di Matematica Universitµa di Roma I, Piazza A. Moro 2, 00185 Roma, Italy
3.	Departamento de Matemáticas, U. Autónoma de Madrid, 28049 Madrid, Spain

Received January 2006 Revised July 2006 Published August 2006

Abstract
Related Papers

We study the effect of a zero order term on existence and optimal summability of solutions to the elliptic problem

$ -\text{div}( M(x)\nabla u)- a\frac{u}{|x|^2}=f \text{ in } \Omega, \qquad u=0 \text{ on } \partial \Omega$,

with respect to the summability of $f$ and the value of the parameter $a$. Here $\Omega$ is a bounded domain in $\mathbb{R}^N$ containing the origin.

Keywords: Laplace equation, Hardy potential summability of solution..

Mathematics Subject Classification: Primary: 35J05, 35J25; Secondary: 46E3.

Citation: Lucio Boccardo, Luigi Orsina, Ireneo Peral. A remark on existence and optimal summability of solutions of elliptic problems involving Hardy potential. Discrete & Continuous Dynamical Systems - A, 2006, 16 (3) : 513-523. doi: 10.3934/dcds.2006.16.513

[1]	Fengshuang Gao, Yuxia Guo. Multiple solutions for a critical quasilinear equation with Hardy potential. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 1977-2003. doi: 10.3934/dcdss.2019128
[2]	Roman Chapko, B. Tomas Johansson. On the numerical solution of a Cauchy problem for the Laplace equation via a direct integral equation approach. Inverse Problems & Imaging, 2012, 6 (1) : 25-38. doi: 10.3934/ipi.2012.6.25
[3]	Yinbin Deng, Qi Gao, Dandan Zhang. Nodal solutions for Laplace equations with critical Sobolev and Hardy exponents on $R^N$. Discrete & Continuous Dynamical Systems - A, 2007, 19 (1) : 211-233. doi: 10.3934/dcds.2007.19.211
[4]	Boumediene Abdellaoui, Ahmed Attar. Quasilinear elliptic problem with Hardy potential and singular term. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1363-1380. doi: 10.3934/cpaa.2013.12.1363
[5]	Jian Zhang, Shihui Zhu, Xiaoguang Li. Rate of $L^2$-concentration of the blow-up solution for critical nonlinear Schrödinger equation with potential. Mathematical Control & Related Fields, 2011, 1 (1) : 119-127. doi: 10.3934/mcrf.2011.1.119
[6]	Guoqing Zhang, Jia-yu Shao, Sanyang Liu. Linking solutions for N-laplace elliptic equations with Hardy-Sobolev operator and indefinite weights. Communications on Pure & Applied Analysis, 2011, 10 (2) : 571-581. doi: 10.3934/cpaa.2011.10.571
[7]	Chao Zhang, Xia Zhang, Shulin Zhou. Gradient estimates for the strong $p(x)$-Laplace equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 4109-4129. doi: 10.3934/dcds.2017175
[8]	Yutian Lei, Congming Li, Chao Ma. Decay estimation for positive solutions of a $\gamma$-Laplace equation. Discrete & Continuous Dynamical Systems - A, 2011, 30 (2) : 547-558. doi: 10.3934/dcds.2011.30.547
[9]	Jann-Long Chern, Yong-Li Tang, Chuan-Jen Chyan, Yi-Jung Chen. On the uniqueness of singular solutions for a Hardy-Sobolev equation. Conference Publications, 2013, 2013 (special) : 123-128. doi: 10.3934/proc.2013.2013.123
[10]	Jing Zhang, Shiwang Ma. Positive solutions of perturbed elliptic problems involving Hardy potential and critical Sobolev exponent. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1999-2009. doi: 10.3934/dcdsb.2016033
[11]	Boumediene Abdellaoui, Daniela Giachetti, Ireneo Peral, Magdalena Walias. Elliptic problems with nonlinear terms depending on the gradient and singular on the boundary: Interaction with a Hardy-Leray potential. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 1747-1774. doi: 10.3934/dcds.2014.34.1747
[12]	Vladimir Georgiev, Sandra Lucente. Focusing nlkg equation with singular potential. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1387-1406. doi: 10.3934/cpaa.2018068
[13]	Zifei Shen, Fashun Gao, Minbo Yang. On critical Choquard equation with potential well. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3567-3593. doi: 10.3934/dcds.2018151
[14]	Samia Challal, Abdeslem Lyaghfouri. Hölder continuity of solutions to the $A$-Laplace equation involving measures. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1577-1583. doi: 10.3934/cpaa.2009.8.1577
[15]	Marek Fila, Kazuhiro Ishige, Tatsuki Kawakami. Convergence to the Poisson kernel for the Laplace equation with a nonlinear dynamical boundary condition. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1285-1301. doi: 10.3934/cpaa.2012.11.1285
[16]	John C. Schotland, Vadim A. Markel. Fourier-Laplace structure of the inverse scattering problem for the radiative transport equation. Inverse Problems & Imaging, 2007, 1 (1) : 181-188. doi: 10.3934/ipi.2007.1.181
[17]	Yinbin Deng, Qi Gao. Asymptotic behavior of the positive solutions for an elliptic equation with Hardy term. Discrete & Continuous Dynamical Systems - A, 2009, 24 (2) : 367-380. doi: 10.3934/dcds.2009.24.367
[18]	Út V. Lê. Regularity of the solution of a nonlinear wave equation. Communications on Pure & Applied Analysis, 2010, 9 (4) : 1099-1115. doi: 10.3934/cpaa.2010.9.1099
[19]	Giuseppe Maria Coclite, Lorenzo di Ruvo. A note on the convergence of the solution of the Novikov equation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2865-2899. doi: 10.3934/dcdsb.2018290
[20]	Xiaolong Han, Guozhen Lu. Regularity of solutions to an integral equation associated with Bessel potential. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1111-1119. doi: 10.3934/cpaa.2011.10.1111

2018 Impact Factor: 1.143

American Institute of Mathematical Sciences