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

Remarks on weak solutions of fractional elliptic equations

Wanwan Wang 1, , Hongxia Zhang 1, and  Huyuan Chen 1,

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 $1+\frac{2\alpha}{N}\not=\frac{N}{2\alpha}$.
Keywords: Green kernel., weak solution, Fractional Laplacian, Dirac mass, asymptotic behavior.
Mathematics Subject Classification: Primary: 35R11, 35J61; Secondary: 35R0.
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:
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W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343. doi: 10.1002/cpa.20116.  Google Scholar

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X. Chen and J. Yang, Limiting behavior of solutions to an equation with the fractional Laplacian, Diff. Integral Equations, 27 (2014), 157-179.  Google Scholar

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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

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M. Marcus and A. C. Ponce, Reduced limits for nonlinear equations with measures, J. Funct. Anal., 258 (2010), 2316-2372. doi: 10.1016/j.jfa.2009.09.007.  Google Scholar

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J. Tan, Y. Wang and J. Yang, Nonlinear fractional field equations, Nonlinear Analysis: Theory, Methods & Applications, 75 (2012), 2098-2110. doi: 10.1016/j.na.2011.10.010.  Google Scholar

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L. Véron, Elliptic equations involving Measures, Stationary Partial Differential Equations, Vol. I, 593-712, Handb. Differ. Eq., North-Holland, Amsterdam, 2004. 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), 673-770. 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, Erice, Wiley, Chichester, (1980), 53-73.  Google Scholar

[3]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Part. Diff. Eq., 32 (2007), 1245-1260. 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), 23-53. 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), 911-941. 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), 1678-1732. 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), 465-501. doi: 10.1007/s002080050232.  Google Scholar

[8]

H. Chen and L. Véron, Semilinear fractional elliptic equations involving measures, J. Diff. Eq., 257 (2014), 1457-1486. 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), 5467-5492. 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), 165-184.  Google Scholar

[11]

H. Chen and J. Yang, Semilinear fractional elliptic equations with measures in unbounded domain, arXiv: 1403.1530. 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), 347-354.  Google Scholar

[13]

W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343. 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), 157-179.  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), 2316-2372. 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, Methods & Applications, 75 (2012), 2098-2110. 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), 225-242. doi: 10.1016/0362-546X(81)90028-6.  Google Scholar

[19]

L. Véron, Elliptic equations involving Measures, Stationary Partial Differential Equations, Vol. I, 593-712, Handb. Differ. Eq., North-Holland, Amsterdam, 2004. doi: 10.1016/S1874-5733(04)80010-X.  Google Scholar

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