# American Institute of Mathematical Sciences

February  2005, 12(2): 347-354. doi: 10.3934/dcds.2005.12.347

## Qualitative properties of solutions for an integral equation

 1 Department of Mathematics, Yeshiva University, 500 W 185th Street, New York, NY 10033, United States 2 Department of Applied Mathematics, University of Colorado at Boulder 3 Department of Mathematics, University of Toledo, Toledo OH 43606

Received  August 2003 Revised  June 2004 Published  December 2004

Let $n$ be a positive integer and let $0 < \alpha < n.$ In this paper, we study more general integral equation

$u(x) = \int_{R^n} \frac{1}{|x-y|^{n-\alpha}} K(y) u(y)^p dy. We establish regularity, radial symmetry, and monotonicity of the solutions. We also consider subcritical cases, super critical cases, and singular solutions in all cases; and obtain qualitative properties for these solutions. Citation: Wenxiong Chen, Congming Li, Biao Ou. Qualitative properties of solutions for an integral equation. Discrete & Continuous Dynamical Systems - A, 2005, 12 (2) : 347-354. doi: 10.3934/dcds.2005.12.347  [1] Yuxia Guo, Shaolong Peng. A direct method of moving planes for fully nonlinear nonlocal operators and applications. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020462 [2] Norman Noguera, Ademir Pastor. Scattering of radial solutions for quadratic-type Schrödinger systems in dimension five. Discrete & Continuous Dynamical Systems - A, 2021 doi: 10.3934/dcds.2021018 [3] Lihong Zhang, Wenwen Hou, Bashir Ahmad, Guotao Wang. Radial symmetry for logarithmic Choquard equation involving a generalized tempered fractional$ p $-Laplacian. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020445 [4] Lucio Damascelli, Filomena Pacella. Sectional symmetry of solutions of elliptic systems in cylindrical domains. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3305-3325. doi: 10.3934/dcds.2020045 [5] Chaman Kumar. On Milstein-type scheme for SDE driven by Lévy noise with super-linear coefficients. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1405-1446. doi: 10.3934/dcdsb.2020167 [6] Qingfang Wang, Hua Yang. Solutions of nonlocal problem with critical exponent. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5591-5608. doi: 10.3934/cpaa.2020253 [7] Kai Yang. Scattering of the focusing energy-critical NLS with inverse square potential in the radial case. Communications on Pure & Applied Analysis, 2021, 20 (1) : 77-99. doi: 10.3934/cpaa.2020258 [8] João Vitor da Silva, Hernán Vivas. Sharp regularity for degenerate obstacle type problems: A geometric approach. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1359-1385. doi: 10.3934/dcds.2020321 [9] Joel Kübler, Tobias Weth. Spectral asymptotics of radial solutions and nonradial bifurcation for the Hénon equation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3629-3656. doi: 10.3934/dcds.2020032 [10] 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 [11] Jens Lorenz, Wilberclay G. Melo, Suelen C. P. de Souza. Regularity criteria for weak solutions of the Magneto-micropolar equations. Electronic Research Archive, 2021, 29 (1) : 1625-1639. doi: 10.3934/era.2020083 [12] Mengni Li. Global regularity for a class of Monge-Ampère type equations with nonzero boundary conditions. Communications on Pure & Applied Analysis, 2021, 20 (1) : 301-317. doi: 10.3934/cpaa.2020267 [13] Zedong Yang, Guotao Wang, Ravi P. Agarwal, Haiyong Xu. Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020436 [14] Li Cai, Fubao Zhang. The Brezis-Nirenberg type double critical problem for a class of Schrödinger-Poisson equations. Electronic Research Archive, , () : -. doi: 10.3934/era.2020125 [15] Manuel del Pino, Monica Musso, Juncheng Wei, Yifu Zhou. Type Ⅱ finite time blow-up for the energy critical heat equation in$ \mathbb{R}^4 \$. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3327-3355. doi: 10.3934/dcds.2020052 [16] Junyong Eom, Kazuhiro Ishige. Large time behavior of ODE type solutions to nonlinear diffusion equations. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3395-3409. doi: 10.3934/dcds.2019229 [17] Makram Hamouda, Ahmed Bchatnia, Mohamed Ali Ayadi. Numerical solutions for a Timoshenko-type system with thermoelasticity with second sound. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021001 [18] Chungen Liu, Huabo Zhang. Ground state and nodal solutions for fractional Schrödinger-maxwell-kirchhoff systems with pure critical growth nonlinearity. Communications on Pure & Applied Analysis, 2021, 20 (2) : 817-834. doi: 10.3934/cpaa.2020292 [19] Lingwei Ma, Zhenqiu Zhang. Monotonicity for fractional Laplacian systems in unbounded Lipschitz domains. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 537-552. doi: 10.3934/dcds.2020268 [20] Gongbao Li, Tao Yang. Improved Sobolev inequalities involving weighted Morrey norms and the existence of nontrivial solutions to doubly critical elliptic systems involving fractional Laplacian and Hardy terms. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020469

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