March  2016, 15(2): 335-340. doi: 10.3934/cpaa.2016.15.335

Remarks on weak solutions of fractional elliptic equations

1. 

Department of Mathematics, Jiangxi Normal University, Nanchang, Jiangxi 330022, China, China

Received  January 2015 Revised  October 2015 Published  January 2016

In this note, we continue our study of weak solution $u_k$ to fractional elliptic equation $(-\Delta)^\alpha u+u^p=k\delta_0$ in $\Omega$ which vanishes in $\Omega^c$, where $\Omega\subset \mathbb{R}^N (N\ge2)$ is an open $C^2$ domain containing $0$, $(-\Delta)^\alpha$ with $\alpha\in(0,1)$ is the fractional Laplacian, $k>0$ and $\delta_0$ is the Dirac mass at $0$. We prove that the limit of $u_k$ as $k\to\infty$ blows up in whole $\Omega$ when $p=\min\{1+\frac{2\alpha}{N},\frac{N}{2\alpha}\}$ and $1+\frac{2\alpha}{N}\not=\frac{N}{2\alpha}$.
Citation: Wanwan Wang, Hongxia Zhang, Huyuan Chen. Remarks on weak solutions of fractional elliptic equations. Communications on Pure & Applied Analysis, 2016, 15 (2) : 335-340. doi: 10.3934/cpaa.2016.15.335
References:
[1]

Ph. Bénilan and H. Brezis, Nonlinear problems related to the Thomas-Fermi equation,, J. Evolution Eq., 3 (2003), 673. doi: 10.1007/s00028-003-0117-8. Google Scholar

[2]

H. Brezis, Some variational problems of the Thomas-Fermi type. Variational inequalities and complementarity problems,, Proc. Internat. School, (1980), 53. Google Scholar

[3]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian,, Comm. Part. Diff. Eq., 32 (2007), 1245. doi: 10.1080/03605300600987306. Google Scholar

[4]

X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates,, In Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 31 (2014), 23. doi: 10.1016/j.anihpc.2013.02.001. Google Scholar

[5]

X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians II: existence, uniqueness, and qualitative properties of solutions,, Trans. American Mathematical Society, 367 (2015), 911. doi: 10.1090/S0002-9947-2014-05906-0. Google Scholar

[6]

X. Cabré and J. Solà-Morales, Layer solutions in a half-space for boundary reactions,, Comm. Pure Appl. Math., 58 (2005), 1678. doi: 10.1002/cpa.20093. Google Scholar

[7]

Z. Chen and R. Song, Estimates on Green functions and poisson kernels for symmetric stable process,, Math. Ann., 312 (1998), 465. doi: 10.1007/s002080050232. Google Scholar

[8]

H. Chen and L. Véron, Semilinear fractional elliptic equations involving measures,, J. Diff. Eq., 257 (2014), 1457. doi: 10.1016/j.jde.2014.05.012. Google Scholar

[9]

H. Chen and L. Véron, Semilinear fractional elliptic equations with gradient nonlinearity involving measures,, J. Funct. Anal., 266 (2014), 5467. doi: 10.1016/j.jfa.2013.11.009. Google Scholar

[10]

H. Chen and L. Véron, Weakly and strongly singular solutions of semilinear fractional elliptic equations,, Asymptotic Analysis, 88 (2014), 165. Google Scholar

[11]

H. Chen and J. Yang, Semilinear fractional elliptic equations with measures in unbounded domain,, arXiv: 1403.1530., (1403). Google Scholar

[12]

W. Chen, C. Li and B. Ou, Qualitative properties of solutions for an integral equation,, Discrete and Continuous Dynamical Systems, 12 (2005), 347. Google Scholar

[13]

W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation,, Comm. Pure Appl. Math., 59 (2006), 330. doi: 10.1002/cpa.20116. Google Scholar

[14]

X. Chen and J. Yang, Limiting behavior of solutions to an equation with the fractional Laplacian,, Diff. Integral Equations, 27 (2014), 157. Google Scholar

[15]

P. Felmer and Y. Wang, Radial symmetry of positive solutions to equations involving the fractional laplacian,, Comm. Cont. Math., 16 (2014). doi: 10.1142/S0219199713500235. Google Scholar

[16]

M. Marcus and A. C. Ponce, Reduced limits for nonlinear equations with measures,, J. Funct. Anal., 258 (2010), 2316. doi: 10.1016/j.jfa.2009.09.007. Google Scholar

[17]

J. Tan, Y. Wang and J. Yang, Nonlinear fractional field equations,, Nonlinear Analysis: Theory, 75 (2012), 2098. doi: 10.1016/j.na.2011.10.010. Google Scholar

[18]

L. Véron, Singular solutions of some nonlinear elliptic equations,, Nonlinear Anal. T. M. $&$ A., 5 (1981), 225. doi: 10.1016/0362-546X(81)90028-6. Google Scholar

[19]

L. Véron, Elliptic equations involving Measures, Stationary Partial Differential Equations,, Vol. I, (2004), 593. doi: 10.1016/S1874-5733(04)80010-X. Google Scholar

show all references

References:
[1]

Ph. Bénilan and H. Brezis, Nonlinear problems related to the Thomas-Fermi equation,, J. Evolution Eq., 3 (2003), 673. doi: 10.1007/s00028-003-0117-8. Google Scholar

[2]

H. Brezis, Some variational problems of the Thomas-Fermi type. Variational inequalities and complementarity problems,, Proc. Internat. School, (1980), 53. Google Scholar

[3]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian,, Comm. Part. Diff. Eq., 32 (2007), 1245. doi: 10.1080/03605300600987306. Google Scholar

[4]

X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates,, In Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 31 (2014), 23. doi: 10.1016/j.anihpc.2013.02.001. Google Scholar

[5]

X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians II: existence, uniqueness, and qualitative properties of solutions,, Trans. American Mathematical Society, 367 (2015), 911. doi: 10.1090/S0002-9947-2014-05906-0. Google Scholar

[6]

X. Cabré and J. Solà-Morales, Layer solutions in a half-space for boundary reactions,, Comm. Pure Appl. Math., 58 (2005), 1678. doi: 10.1002/cpa.20093. Google Scholar

[7]

Z. Chen and R. Song, Estimates on Green functions and poisson kernels for symmetric stable process,, Math. Ann., 312 (1998), 465. doi: 10.1007/s002080050232. Google Scholar

[8]

H. Chen and L. Véron, Semilinear fractional elliptic equations involving measures,, J. Diff. Eq., 257 (2014), 1457. doi: 10.1016/j.jde.2014.05.012. Google Scholar

[9]

H. Chen and L. Véron, Semilinear fractional elliptic equations with gradient nonlinearity involving measures,, J. Funct. Anal., 266 (2014), 5467. doi: 10.1016/j.jfa.2013.11.009. Google Scholar

[10]

H. Chen and L. Véron, Weakly and strongly singular solutions of semilinear fractional elliptic equations,, Asymptotic Analysis, 88 (2014), 165. Google Scholar

[11]

H. Chen and J. Yang, Semilinear fractional elliptic equations with measures in unbounded domain,, arXiv: 1403.1530., (1403). Google Scholar

[12]

W. Chen, C. Li and B. Ou, Qualitative properties of solutions for an integral equation,, Discrete and Continuous Dynamical Systems, 12 (2005), 347. Google Scholar

[13]

W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation,, Comm. Pure Appl. Math., 59 (2006), 330. doi: 10.1002/cpa.20116. Google Scholar

[14]

X. Chen and J. Yang, Limiting behavior of solutions to an equation with the fractional Laplacian,, Diff. Integral Equations, 27 (2014), 157. Google Scholar

[15]

P. Felmer and Y. Wang, Radial symmetry of positive solutions to equations involving the fractional laplacian,, Comm. Cont. Math., 16 (2014). doi: 10.1142/S0219199713500235. Google Scholar

[16]

M. Marcus and A. C. Ponce, Reduced limits for nonlinear equations with measures,, J. Funct. Anal., 258 (2010), 2316. doi: 10.1016/j.jfa.2009.09.007. Google Scholar

[17]

J. Tan, Y. Wang and J. Yang, Nonlinear fractional field equations,, Nonlinear Analysis: Theory, 75 (2012), 2098. doi: 10.1016/j.na.2011.10.010. Google Scholar

[18]

L. Véron, Singular solutions of some nonlinear elliptic equations,, Nonlinear Anal. T. M. $&$ A., 5 (1981), 225. doi: 10.1016/0362-546X(81)90028-6. Google Scholar

[19]

L. Véron, Elliptic equations involving Measures, Stationary Partial Differential Equations,, Vol. I, (2004), 593. doi: 10.1016/S1874-5733(04)80010-X. Google Scholar

[1]

Xi Wang, Zuhan Liu, Ling Zhou. Asymptotic decay for the classical solution of the chemotaxis system with fractional Laplacian in high dimensions. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 4003-4020. doi: 10.3934/dcdsb.2018121

[2]

Bernard Brighi, S. Guesmia. Asymptotic behavior of solution of hyperbolic problems on a cylindrical domain. Conference Publications, 2007, 2007 (Special) : 160-169. doi: 10.3934/proc.2007.2007.160

[3]

Everaldo S. de Medeiros, Jianfu Yang. Asymptotic behavior of solutions to a perturbed p-Laplacian problem with Neumann condition. Discrete & Continuous Dynamical Systems - A, 2005, 12 (4) : 595-606. doi: 10.3934/dcds.2005.12.595

[4]

Belkacem Said-Houari, Radouane Rahali. Asymptotic behavior of the solution to the Cauchy problem for the Timoshenko system in thermoelasticity of type III. Evolution Equations & Control Theory, 2013, 2 (2) : 423-440. doi: 10.3934/eect.2013.2.423

[5]

Ling Mi. Asymptotic behavior for the unique positive solution to a singular elliptic problem. Communications on Pure & Applied Analysis, 2015, 14 (3) : 1053-1072. doi: 10.3934/cpaa.2015.14.1053

[6]

Bhargav Kumar Kakumani, Suman Kumar Tumuluri. Asymptotic behavior of the solution of a diffusion equation with nonlocal boundary conditions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 407-419. doi: 10.3934/dcdsb.2017019

[7]

Tingting Liu, Qiaozhen Ma. Time-dependent asymptotic behavior of the solution for plate equations with linear memory. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4595-4616. doi: 10.3934/dcdsb.2018178

[8]

Ciprian G. Gal, Hao Wu. Asymptotic behavior of a Cahn-Hilliard equation with Wentzell boundary conditions and mass conservation. Discrete & Continuous Dynamical Systems - A, 2008, 22 (4) : 1041-1063. doi: 10.3934/dcds.2008.22.1041

[9]

Zhong Tan, Zheng-An Yao. The existence and asymptotic behavior of the evolution p-Laplacian equations with strong nonlinear sources. Communications on Pure & Applied Analysis, 2004, 3 (3) : 475-490. doi: 10.3934/cpaa.2004.3.475

[10]

Na An, Chaobao Huang, Xijun Yu. Error analysis of discontinuous Galerkin method for the time fractional KdV equation with weak singularity solution. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2019185

[11]

Luyi Ma, Hong-Tao Niu, Zhi-Cheng Wang. Global asymptotic stability of traveling waves to the Allen-Cahn equation with a fractional Laplacian. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2457-2472. doi: 10.3934/cpaa.2019111

[12]

Guofu Lu. Nonexistence and short time asymptotic behavior of source-type solution for porous medium equation with convection in one-dimension. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1567-1586. doi: 10.3934/dcdsb.2016011

[13]

Jean-Claude Saut, Jun-Ichi Segata. Asymptotic behavior in time of solution to the nonlinear Schrödinger equation with higher order anisotropic dispersion. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 219-239. doi: 10.3934/dcds.2019009

[14]

Yutong Chen, Jiabao Su. Resonant problems for fractional Laplacian. Communications on Pure & Applied Analysis, 2017, 16 (1) : 163-188. doi: 10.3934/cpaa.2017008

[15]

Philip M. J. Trevelyan. Approximating the large time asymptotic reaction zone solution for fractional order kinetics $A^n B^m$. Discrete & Continuous Dynamical Systems - S, 2012, 5 (1) : 219-234. doi: 10.3934/dcdss.2012.5.219

[16]

Bo-Qing Dong, Juan Song. Global regularity and asymptotic behavior of modified Navier-Stokes equations with fractional dissipation. Discrete & Continuous Dynamical Systems - A, 2012, 32 (1) : 57-79. doi: 10.3934/dcds.2012.32.57

[17]

Claudia Bucur. Some observations on the Green function for the ball in the fractional Laplace framework. Communications on Pure & Applied Analysis, 2016, 15 (2) : 657-699. doi: 10.3934/cpaa.2016.15.657

[18]

Lei Qiao. Asymptotic behaviors of Green-Sch potentials at infinity and its applications. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2321-2338. doi: 10.3934/dcdsb.2017099

[19]

Federico Cacciafesta, Anne-Sophie De Suzzoni. Weak dispersion for the Dirac equation on asymptotically flat and warped product spaces. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4359-4398. doi: 10.3934/dcds.2019177

[20]

Maoding Zhen, Jinchun He, Haoyun Xu. Critical system involving fractional Laplacian. Communications on Pure & Applied Analysis, 2019, 18 (1) : 237-253. doi: 10.3934/cpaa.2019013

2018 Impact Factor: 0.925

Metrics

  • PDF downloads (9)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]