Remarks on weak solutions of fractional elliptic equations

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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

Abstract
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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 and Applied Analysis, 2016, 15 (2) : 335-340. doi: 10.3934/cpaa.2016.15.335

References:

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[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.
[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.
[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.
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[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.
[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.
[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.
[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.
[10]	H. Chen and L. Véron, Weakly and strongly singular solutions of semilinear fractional elliptic equations, Asymptotic Analysis, 88 (2014), 165-184.
[11]	H. Chen and J. Yang, Semilinear fractional elliptic equations with measures in unbounded domain, arXiv: 1403.1530.
[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.
[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.
[14]	X. Chen and J. Yang, Limiting behavior of solutions to an equation with the fractional Laplacian, Diff. Integral Equations, 27 (2014), 157-179.
[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.
[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.
[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.
[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.
[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.

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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[10]	H. Chen and L. Véron, Weakly and strongly singular solutions of semilinear fractional elliptic equations, Asymptotic Analysis, 88 (2014), 165-184.
[11]	H. Chen and J. Yang, Semilinear fractional elliptic equations with measures in unbounded domain, arXiv: 1403.1530.
[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.
[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.
[14]	X. Chen and J. Yang, Limiting behavior of solutions to an equation with the fractional Laplacian, Diff. Integral Equations, 27 (2014), 157-179.
[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.
[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.
[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.
[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.
[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.

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