# American Institute of Mathematical Sciences

November  2015, 14(6): 2393-2409. doi: 10.3934/cpaa.2015.14.2393

## Symmetry of solutions to semilinear equations involving the fractional laplacian

 1 School of Mathematics and Information Science, Henan Normal University, Xinxiang, 453007

Received  February 2015 Revised  July 2015 Published  September 2015

Let $0<\alpha<2$ be any real number. Let $\Omega\subset\mathbb{R}^n$ be a bounded or an unbounded domain which is convex and symmetric in $x_1$ direction. We investigate the following Dirichlet problem involving the fractional Laplacian: $$\left\{\begin{array}{ll} (-\Delta)^{\alpha/2} u(x)=f(x,u), & \qquad x\in\Omega, (1)\\ u(x)\equiv0, & \qquad x\notin\Omega. \end{array}\right.$$
Applying a direct method of moving planes for the fractional Laplacian, with the help of several maximum principles for anti-symmetric functions, we prove the monotonicity and symmetry of positive solutions in $x_1$ direction as well as nonexistence of positive solutions under various conditions on $f$ and on the solutions $u$. We also extend the results to some more complicated cases.
Citation: Lizhi Zhang. Symmetry of solutions to semilinear equations involving the fractional laplacian. Communications on Pure and Applied Analysis, 2015, 14 (6) : 2393-2409. doi: 10.3934/cpaa.2015.14.2393
##### References:
 [1] F. V. Atkinson and L. A. Peletier, Elliptic equations with nearly critical growth, J. Diff. Equ., 70 (1987), 349-365. doi: 10.1016/0022-0396(87)90156-2. [2] H. Berestycki, L. Caffarelli and L. Nirenberg, Symmetry for elliptic equations in a halfspace, in Boundary Value Problems for Partial Differential Equations and Applications (eds. volume dedicated to E. Magenes, J. L. Lions et al.), Masson, Paris, (1993), 27-42. [3] H. Berestycki, L. Caffarelli and L. Nirenberg, Inequalities for second order elliptic equations with applications to unbounded domains I, Duke Math. J., 81 (1996), 467-494. doi: 10.1215/S0012-7094-96-08117-X. [4] H. Berestycki, L. Caffarelli and L. Nirenberg, Monotonicity for elliptic equations in unbounded Lipschitz domains, Comm. Pure Appl. Math., 50 (1997), 1089-1111. doi: 10.1002/(SICI)1097-0312(199711)50:11<1089::AID-CPA2>3.0.CO;2-6. [5] H. Berestycki and L. Nirenberg, On the method of moving planes and the sliding method, Bol. Soc. Brasil Mat. (N.S.), 22 (1991), 1-37. doi: 10.1007/BF01244896. [6] H. Berestycki and L. Nirenberg, Monotonicity, symmetry and antisymmetry of solutions of semilinear elliptic equations, J. Geom. and Physics, 5 (1988), 237-275. doi: 10.1016/0393-0440(88)90006-X. [7] C. Brandle, E. Colorado, A. de Pablo and U. Sanchez, A concave-convex elliptic problem involving the fractional Laplacian, Proc. Royal Soc. Edinburgh, 143 (2013), 39-71. doi: 10.1017/S0308210511000175. [8] H. Brezis and L. A. Peletier, Asymptotics for elliptic equations involving critical growth, in Partial Differential Equations and the Calculus of Variations (Progr. Nonlinear Differential Equations Appl., 1), Birkhäuser Boston, (1989), 149-192. [9] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260. doi: 10.1080/03605300600987306. [10] L. Cao and W. Chen, Liouville type theorems for poly-harmonic Navier problems, Discrete Contin. Dyn. Sys., 33 (2013), 3937-3955. doi: 10.3934/dcds.2013.33.3937. [11] W. Chen, Y. Fang and C. Li, Super poly-harmonic property of solutions for Navier boundary problems on a half space, J. Funct. Anal., 256 (2013), 1522-1555. doi: 10.1016/j.jfa.2013.06.010. [12] W. Chen, Y. Fang and R. Yang, Liouville theorems involving the fractional Laplacian on a half space, Advances in Math., 274 (2015), 167-198. doi: 10.1016/j.aim.2014.12.013. [13] W. Chen and C. Li, Regularity of solutions for a system of integral equation, Comm. Pure Appl. Anal., 4 (2005), 1-8. [14] W. Chen and C. Li, Radial symmetry of solutions for some integral systems of Wolff type, Disc. Cont. Dyn. Sys., 30 (2011), 1083-1093. doi: 10.3934/dcds.2011.30.1083. [15] W. Chen, C. Li and Y. Li, A direct method of moving planes for the fractional Laplacian, preprint, arXiv:1411.1697. [16] 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. [17] D. Chen and L. Ma, A Liouville type theorem for an integral system, Comm. Pure Appl. Anal., 5 (2006), 855-859. doi: 10.3934/cpaa.2006.5.855. [18] W. Chen and J. Zhu, Radial symmetry and regularity of solutions for poly-harmonic Dirichlet problems, J. Math. Anal. Appl., 377 (2011), 744-753. doi: 10.1016/j.jmaa.2010.11.035. [19] C. V. Coffman, Uniqueness of the ground state solution for $\Delta u-u+u^3$ and a variational characterization of other solutions, Arch. Rational Mech. Anal., 46 (1972), 81-95. doi: 10.1007/BF00250684. [20] M. M. Fall and T. Weth, Monotonicity and nonexistence results for some fractional elliptic problems in the half space, preprint, arXiv:1309.7230. [21] Y. Fang and W. Chen, A Liouville type theorem for poly-harmonic Dirichlet problems in a half space, Advances in Math., 229 (2012), 2835-2867. doi: 10.1016/j.aim.2012.01.018. [22] D. G. Figueiredo, P. L. Lions and R. D. Nussbaum, A priori estimates and existence of positive solutions of semilinear elliptic equations, J. Math. Pures et Appl., 61 (1982), 41-63. [23] R. L. Frank, E. Lenzmann and L. Silvestre, Uniqueness of radial solutions for the fractional Laplacian, preprint, arXiv:1302.2652. [24] B. Gidas, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243. doi: 10.1007/BF01221125. [25] B. Gidas, W. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\mathbb{R}^{N}$, in Math. Anal. and Applications, Part A, Advances in Math. Suppl. Studies 7A (ed. L. Nachbin), Academic Pr., (1981), 369-402. [26] S. Jarohs and Tobias Weth, Symmetry via antisymmetric maximum principles in nonlocal problems of variable order, preprint, arXiv:1406.6181. [27] H. G. Kaper and M. K. Kwong, Uniqueness of non-negative solutions of a class of semilinear elliptic equations, Nonlinear Diffusion Equations and Their Equilibrium States II, 13 (1988), 1-17. [28] C. Li, Monotonicity and symmetry of solutions of fully nonlinear elliptic equations on bounded domains, Commun. in Partial Differential Equations, 16 (1991), 491-526. doi: 10.1080/03605309108820766. [29] C. Li, Monotonicity and symmetry of solutions of fully nonlinear elliptic equations on unbounded domains, Commun. in Partial Differential Equations, 16 (1991), 585-615. doi: 10.1080/03605309108820770. [30] C. Liu and S. Qiao, Symmetry and monotonicity for a system of integral equations, Commun. Pure Appl. Anal., 8 (2009), 1925-1932. doi: 10.3934/cpaa.2009.8.1925. [31] M. K. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p=0$ in $\mathbb{R}^{N}$, Archive for Rational Mechanics and Analysis, 105 (1989), 243-266. doi: 10.1007/BF00251502. [32] K. Mcleod and J. Serrin, Uniqueness of positive radial solutions of $\Delta u+f(u)=0$ in $\mathbb{R}^{N}$, Arch. Rational Mech. Anal., 99 (1987), 115-145. doi: 10.1007/BF00275874. [33] L. A. Peletier and J. Serrin, Uniqueness of positive solutions of semilinear equations in $\mathbb{R}^{N}$, Arch. Rational Mech. Anal., 81 (1983), 181-197. doi: 10.1007/BF00250651. [34] L. Zhang and T. Cheng, Liouville theorems involving the fractional Laplacian on the upper half Euclidean space, submitted to Acta Applicandae Mathematicae. [35] R. Zhuo, W. Chen, X. Cui and Z. Yuan, A Liouville theorem for the fractional Laplacian, Disc. Cont. Dyn. Sys., 36 (2016), arXiv:1401.7402. [36] S. Zhu, X. Chen and J. Yang, Regularity, symmetry and uniqueness of positive solutions to a nonlinear elliptic system, Commun. Pure Appl. Anal., 12 (2013), 2685-2696. doi: 10.3934/cpaa.2013.12.2685. [37] L. Zhang, C. Li, W. Chen and T. Cheng, A Liouville theorem for $\alpha$-harmonic functions in $\mathbbR^n_+$, Disc. Cont. Dyn. Sys., 36 (2016), arXiv:1409.4106. [38] R. Zhuo, F. Li and B. Lv, Liouville type theorems for Schrödinger system with Navier boundary conditions in a half space, Commun. Pure Appl. Anal., 13 (2014), 977-990. doi: 10.3934/cpaa.2014.13.977.

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##### References:
 [1] F. V. Atkinson and L. A. Peletier, Elliptic equations with nearly critical growth, J. Diff. Equ., 70 (1987), 349-365. doi: 10.1016/0022-0396(87)90156-2. [2] H. Berestycki, L. Caffarelli and L. Nirenberg, Symmetry for elliptic equations in a halfspace, in Boundary Value Problems for Partial Differential Equations and Applications (eds. volume dedicated to E. Magenes, J. L. Lions et al.), Masson, Paris, (1993), 27-42. [3] H. Berestycki, L. Caffarelli and L. Nirenberg, Inequalities for second order elliptic equations with applications to unbounded domains I, Duke Math. J., 81 (1996), 467-494. doi: 10.1215/S0012-7094-96-08117-X. [4] H. Berestycki, L. Caffarelli and L. Nirenberg, Monotonicity for elliptic equations in unbounded Lipschitz domains, Comm. Pure Appl. Math., 50 (1997), 1089-1111. doi: 10.1002/(SICI)1097-0312(199711)50:11<1089::AID-CPA2>3.0.CO;2-6. [5] H. Berestycki and L. Nirenberg, On the method of moving planes and the sliding method, Bol. Soc. Brasil Mat. (N.S.), 22 (1991), 1-37. doi: 10.1007/BF01244896. [6] H. Berestycki and L. Nirenberg, Monotonicity, symmetry and antisymmetry of solutions of semilinear elliptic equations, J. Geom. and Physics, 5 (1988), 237-275. doi: 10.1016/0393-0440(88)90006-X. [7] C. Brandle, E. Colorado, A. de Pablo and U. Sanchez, A concave-convex elliptic problem involving the fractional Laplacian, Proc. Royal Soc. Edinburgh, 143 (2013), 39-71. doi: 10.1017/S0308210511000175. [8] H. Brezis and L. A. Peletier, Asymptotics for elliptic equations involving critical growth, in Partial Differential Equations and the Calculus of Variations (Progr. Nonlinear Differential Equations Appl., 1), Birkhäuser Boston, (1989), 149-192. [9] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260. doi: 10.1080/03605300600987306. [10] L. Cao and W. Chen, Liouville type theorems for poly-harmonic Navier problems, Discrete Contin. Dyn. Sys., 33 (2013), 3937-3955. doi: 10.3934/dcds.2013.33.3937. [11] W. Chen, Y. Fang and C. Li, Super poly-harmonic property of solutions for Navier boundary problems on a half space, J. Funct. Anal., 256 (2013), 1522-1555. doi: 10.1016/j.jfa.2013.06.010. [12] W. Chen, Y. Fang and R. Yang, Liouville theorems involving the fractional Laplacian on a half space, Advances in Math., 274 (2015), 167-198. doi: 10.1016/j.aim.2014.12.013. [13] W. Chen and C. Li, Regularity of solutions for a system of integral equation, Comm. Pure Appl. Anal., 4 (2005), 1-8. [14] W. Chen and C. Li, Radial symmetry of solutions for some integral systems of Wolff type, Disc. Cont. Dyn. Sys., 30 (2011), 1083-1093. doi: 10.3934/dcds.2011.30.1083. [15] W. Chen, C. Li and Y. Li, A direct method of moving planes for the fractional Laplacian, preprint, arXiv:1411.1697. [16] 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. [17] D. Chen and L. Ma, A Liouville type theorem for an integral system, Comm. Pure Appl. Anal., 5 (2006), 855-859. doi: 10.3934/cpaa.2006.5.855. [18] W. Chen and J. Zhu, Radial symmetry and regularity of solutions for poly-harmonic Dirichlet problems, J. Math. Anal. Appl., 377 (2011), 744-753. doi: 10.1016/j.jmaa.2010.11.035. [19] C. V. Coffman, Uniqueness of the ground state solution for $\Delta u-u+u^3$ and a variational characterization of other solutions, Arch. Rational Mech. Anal., 46 (1972), 81-95. doi: 10.1007/BF00250684. [20] M. M. Fall and T. Weth, Monotonicity and nonexistence results for some fractional elliptic problems in the half space, preprint, arXiv:1309.7230. [21] Y. Fang and W. Chen, A Liouville type theorem for poly-harmonic Dirichlet problems in a half space, Advances in Math., 229 (2012), 2835-2867. doi: 10.1016/j.aim.2012.01.018. [22] D. G. Figueiredo, P. L. Lions and R. D. Nussbaum, A priori estimates and existence of positive solutions of semilinear elliptic equations, J. Math. Pures et Appl., 61 (1982), 41-63. [23] R. L. Frank, E. Lenzmann and L. Silvestre, Uniqueness of radial solutions for the fractional Laplacian, preprint, arXiv:1302.2652. [24] B. Gidas, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243. doi: 10.1007/BF01221125. [25] B. Gidas, W. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\mathbb{R}^{N}$, in Math. Anal. and Applications, Part A, Advances in Math. Suppl. Studies 7A (ed. L. Nachbin), Academic Pr., (1981), 369-402. [26] S. Jarohs and Tobias Weth, Symmetry via antisymmetric maximum principles in nonlocal problems of variable order, preprint, arXiv:1406.6181. [27] H. G. Kaper and M. K. Kwong, Uniqueness of non-negative solutions of a class of semilinear elliptic equations, Nonlinear Diffusion Equations and Their Equilibrium States II, 13 (1988), 1-17. [28] C. Li, Monotonicity and symmetry of solutions of fully nonlinear elliptic equations on bounded domains, Commun. in Partial Differential Equations, 16 (1991), 491-526. doi: 10.1080/03605309108820766. [29] C. Li, Monotonicity and symmetry of solutions of fully nonlinear elliptic equations on unbounded domains, Commun. in Partial Differential Equations, 16 (1991), 585-615. doi: 10.1080/03605309108820770. [30] C. Liu and S. Qiao, Symmetry and monotonicity for a system of integral equations, Commun. Pure Appl. Anal., 8 (2009), 1925-1932. doi: 10.3934/cpaa.2009.8.1925. [31] M. K. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p=0$ in $\mathbb{R}^{N}$, Archive for Rational Mechanics and Analysis, 105 (1989), 243-266. doi: 10.1007/BF00251502. [32] K. Mcleod and J. Serrin, Uniqueness of positive radial solutions of $\Delta u+f(u)=0$ in $\mathbb{R}^{N}$, Arch. Rational Mech. Anal., 99 (1987), 115-145. doi: 10.1007/BF00275874. [33] L. A. Peletier and J. Serrin, Uniqueness of positive solutions of semilinear equations in $\mathbb{R}^{N}$, Arch. Rational Mech. Anal., 81 (1983), 181-197. doi: 10.1007/BF00250651. [34] L. Zhang and T. Cheng, Liouville theorems involving the fractional Laplacian on the upper half Euclidean space, submitted to Acta Applicandae Mathematicae. [35] R. Zhuo, W. Chen, X. Cui and Z. Yuan, A Liouville theorem for the fractional Laplacian, Disc. Cont. Dyn. Sys., 36 (2016), arXiv:1401.7402. [36] S. Zhu, X. Chen and J. Yang, Regularity, symmetry and uniqueness of positive solutions to a nonlinear elliptic system, Commun. Pure Appl. Anal., 12 (2013), 2685-2696. doi: 10.3934/cpaa.2013.12.2685. [37] L. Zhang, C. Li, W. Chen and T. Cheng, A Liouville theorem for $\alpha$-harmonic functions in $\mathbbR^n_+$, Disc. Cont. Dyn. Sys., 36 (2016), arXiv:1409.4106. [38] R. Zhuo, F. Li and B. Lv, Liouville type theorems for Schrödinger system with Navier boundary conditions in a half space, Commun. Pure Appl. Anal., 13 (2014), 977-990. doi: 10.3934/cpaa.2014.13.977.
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