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January  2020, 40(1): 107-131. doi: 10.3934/dcds.2020005

## Regularity of extremal solutions of nonlocal elliptic systems

 Department of Mathematics, The University of Texas at San Antonio, San Antonio, TX 78249, USA

Received  November 2018 Revised  August 2019 Published  October 2019

We examine regularity of the extremal solution of nonlinear nonlocal eigenvalue problem
 $\begin{eqnarray*} \left\{ \begin{array}{lcl} \hfill \mathcal L u & = & \lambda F(u,v) \qquad \text{in} \ \ \Omega, \\ \hfill \mathcal L v & = & \gamma G(u,v) \qquad \text{in} \ \ \Omega, \\ \hfill u,v & = &0 \qquad \qquad \text{on} \ \ \mathbb R^{n} \backslash \Omega , \end{array}\right. \end{eqnarray*}$
with an integro-differential operator, including the fractional Laplacian, of the form
 $\begin{equation*} \label{} \mathcal L(u (x)) = \lim\limits_{\epsilon\to 0} \int_{\mathbb R^n\setminus B_\epsilon(x) } [u(x) - u(z)] J(z-x) dz , \end{equation*}$
when
 $J$
is a nonnegative measurable even jump kernel. In particular, we consider jump kernels of the form of
 $J(y) = \frac{a(y/|y|)}{|y|^{n+2s}}$
where
 $s\in (0,1)$
and
 $a$
is any nonnegative even measurable function in
 $L^1(\mathbb {S}^{n-1})$
that satisfies ellipticity assumptions. We first establish stability inequalities for minimal solutions of the above system for a general nonlinearity and a general kernel. Then, we prove regularity of the extremal solution in dimensions
 $n < 10s$
and
 $n<2s+\frac{4s}{p\mp 1}[p+\sqrt{p(p\mp1)}]$
for the Gelfand and Lane-Emden systems when
 $p>1$
(with positive and negative exponents), respectively. When
 $s\to 1$
, these dimensions are optimal. However, for the case of
 $s\in(0,1)$
getting the optimal dimension remains as an open problem. Moreover, for general nonlinearities, we consider gradient systems and we establish regularity of the extremal solution in dimensions
 $n<4s$
. As far as we know, this is the first regularity result on the extremal solution of nonlocal system of equations.
Citation: Mostafa Fazly. Regularity of extremal solutions of nonlocal elliptic systems. Discrete and Continuous Dynamical Systems, 2020, 40 (1) : 107-131. doi: 10.3934/dcds.2020005
##### References:
 [1] R. Bass, Diffusions and Elliptic Operators, Probability and its Applications, Springer-Verlag, New York, 1998. [2] J. Bertoin, Lévy Processes, Cambridge University Press, Cambridge, 1996. [3] H. Brezis, T. Cazenave, Y. Martel and A. Ramiandrisoa, Blow up for $u_t - \Delta u = g(u)$ revisited, Advances in Differential Equations, 1 (1996), 73-90. [4] H. Brezis and L. Vazquez, Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Complut. Madrid, 10 (1997), 443-469. [5] X. Cabré, Regularity of minimizers of semilinear elliptic problems up to dimension four, Comm. Pure Appl. Math., 63 (2010), 1362-1380.  doi: 10.1002/cpa.20327. [6] X. Cabré and A. Capella, Regularity of radial minimizers and extremal solutions of semilinear elliptic equations, Journal of Functional Analysis, 238 (2006), 709-733.  doi: 10.1016/j.jfa.2005.12.018. [7] X. Cabré, A. Figalli, X. Ros-Oton and J. Serra, Stable solutions to semilinear elliptic equations are smooth up to dimension 9, Preprint, arXiv: 1907.09403 (2019). [8] X. Cabré and X. Ros-Oton, Regularity of stable solutions up to dimension 7 in domains of double revolution, Comm. Partial Differential Equations, 38 (2013), 135-154.  doi: 10.1080/03605302.2012.697505. [9] X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians, Ⅰ: Regularity, maximum principles, and Hamiltonian estimates, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 23-53.  doi: 10.1016/j.anihpc.2013.02.001. [10] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Diff. Equ., 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306. [11] A. Capella, J. Davila, L. Dupaigne and Y. Sire, Regularity of radial extremal solutions for some non local semilinear equations, Comm. Partial Diff. Equ., 36 (2011), 1353-1384.  doi: 10.1080/03605302.2011.562954. [12] W. Chen, C. Li and B. Ou, Classification of solutions to an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343.  doi: 10.1002/cpa.20116. [13] C. Cowan, Regularity of the extremal solutions in a Gelfand system problem, Advanced Nonlinear Studies, 11 (2011), 695-700.  doi: 10.1515/ans-2011-0310. [14] C. Cowan and M. Fazly, Regularity of the extremal solutions associated to elliptic systems, Journal of Differential Equations, 257 (2014), 4087-4107.  doi: 10.1016/j.jde.2014.08.002. [15] M. G. Crandall and P. H. Rabinowitz, Some continuation and variation methods for positive solutions of nonlinear elliptic eigenvalue problems, Arch. Rat. Mech. Anal., 58 (1975), 207-218.  doi: 10.1007/BF00280741. [16] J. Davila, L. Dupaigne and J. Wei, On the fractional Lane-Emden equation, Trans. Amer. Math. Soc., 369 (2017), 6087-6104.  doi: 10.1090/tran/6872. [17] L. Dupaigne, A. Farina and B. Sirakov, Regularity of the extremal solutions for the Liouville system, Geometric Partial Differential Equations, CRM Series, Ed. Norm., Pisa., 15 (2013), 139–144. doi: 10.1007/978-88-7642-473-1_7. [18] P. Esposito, N. Ghoussoub and Y. Guo, Compactness along the branch of semi-stable and unstable solutions for an elliptic problem with a singular nonlinearity, Comm. Pure Appl. Math., 60 (2007), 1731-1768.  doi: 10.1002/cpa.20189. [19] P. Esposito, N. Ghoussoub and Y. Guo, Mathematical Analysis of Partial Differential Equations Modeling Electrostatic MEMS, Research Monograph, Courant Lecture Notes, 2010. doi: 10.1090/cln/020. [20] A. Farina, On the classification of solutions of the Lane-Emden equation on unbounded domains of $\mathbb R^n$, J. Math. Pures Appl., 87 (2007), 537-561.  doi: 10.1016/j.matpur.2007.03.001. [21] A. Farina, Stable solutions of $-\Delta u = e^u$ on $\mathbb R^n$, C. R. Math. Acad. Sci. Paris, 345 (2007), 63-66.  doi: 10.1016/j.crma.2007.05.021. [22] M. Fazly, Rigidity results for stable solutions of symmetric systems, Proceedings of the American Mathematical Society, 143 (2015), 5307-5321.  doi: 10.1090/proc/12647. [23] M. Fazly and N. Ghoussoub, De Giorgi type results for elliptic systems, Calc. Var. Partial Differential Equations, 47 (2013), 809-823.  doi: 10.1007/s00526-012-0536-x. [24] M. Fazly and J. Wei, On finite Morse index solutions of higher order fractional Lane-Emden equations, American Journal of Mathematics, 139 (2017), 433-460.  doi: 10.1353/ajm.2017.0011. [25] X. Fernández-Real and X. Ros-Oton, Boundary regularity for the fractional heat equation, Rev. Real Acad. Cienc. Ser. A Math, 110 (2016), 49-64.  doi: 10.1007/s13398-015-0218-6. [26] N. Ghoussoub and Y. Guo, On the partial differential equations of electro MEMS devices: stationary case, SIAM J. Math. Anal., 38 (2007), 1423-1449.  doi: 10.1137/050647803. [27] P. Glowacki and W. Hebisch, Pointwise estimates for densities of stable semigroups of measures, Studia Math., 104 (1992), 243-258.  doi: 10.4064/sm-104-3-243-258. [28] D. D. Joseph and T. S. Lundgren, Quasilinear Dirichlet problems driven by positive sources, Arch. Rational Mech. Anal., 49 (1973), 241-269.  doi: 10.1007/BF00250508. [29] Ph. Laurencot and C. Walker, Some singular equations modeling MEMS, Bulletin of the American Mathematical Society, 54 (2017), 437–479. doi: 10.1090/bull/1563. [30] Y. Y. Li, Remark on some conformally invariant integral equations: The method of moving spheres, J. Eur. Math. Soc. (JEMS), 6 (2004), 153-180. [31] J. Liouville, Sur l'équation aux différences partielles $\frac{d^2\log \lambda}{du dv}\pm \frac{\lambda}{2a^2}$ = 0, J. Math. Pures Appl., 18 (1853), 71-72. [32] F. Mignot and J. P. Puel, Solution radiale singuliére $-\Delta u = e^u$, C. R. Acad. Sci. Paris, 307 (1988), 379-382. [33] M. Montenegro, Minimal solutions for a class of elliptic systems, Bull. London Math. Soc., 37 (2005), 405-416.  doi: 10.1112/S0024609305004248. [34] K. Nagasaki and T. Suzuki, Spectral and related properties about the Emden-Fowler equation $-\Delta u = \lambda e^u$ on circular domains, Mathematische Annalen, 299 (1994), 1-15.  doi: 10.1007/BF01459770. [35] G. Nedev, Regularity of the extremal solution of semilinear elliptic equations, C. R. Acad. Sci. Paris. I Math., 330 (2000), 997-1002.  doi: 10.1016/S0764-4442(00)00289-5. [36] X. Ros-Oton and J. Serra, The extremal solution for the fractional Laplacian, Calc. Var. Partial Differential Equations, 50 (2014), 723-750.  doi: 10.1007/s00526-013-0653-1. [37] X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary, J. Math. Pures Appl., 101 (2014), 275-302.  doi: 10.1016/j.matpur.2013.06.003. [38] X. Ros-Oton and J. Serra, Regularity theory for general stable operators, Journal Differential Equations, 260 (2016), 8675-8715.  doi: 10.1016/j.jde.2016.02.033. [39] X. Ros-Oton, Regularity for the fractional Gelfand problem up to dimension 7, J. Math. Anal. Appl., 419 (2014), 10-19.  doi: 10.1016/j.jmaa.2014.04.048. [40] T. Sanz-Perela, Regularity of radial stable solutions to semilinear elliptic equations for the fractional Laplacian, Commun. Pure Appl. Anal., 17 (2018), 2547-2575.  doi: 10.3934/cpaa.2018121. [41] L. Silvestre, Hölder estimates for solutions of integro-differential equations like the fractional Laplace, Indiana University Mathematics Journal, 55 (2006), 1155-1174.  doi: 10.1512/iumj.2006.55.2706. [42] E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, vol. 30. Princeton University Press, Princeton, 1970. [43] S. Villegas, Boundedness of extremal solutions in dimension 4, Advances Math., 235 (2013), 126-133.  doi: 10.1016/j.aim.2012.11.015.

show all references

##### References:
 [1] R. Bass, Diffusions and Elliptic Operators, Probability and its Applications, Springer-Verlag, New York, 1998. [2] J. Bertoin, Lévy Processes, Cambridge University Press, Cambridge, 1996. [3] H. Brezis, T. Cazenave, Y. Martel and A. Ramiandrisoa, Blow up for $u_t - \Delta u = g(u)$ revisited, Advances in Differential Equations, 1 (1996), 73-90. [4] H. Brezis and L. Vazquez, Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Complut. Madrid, 10 (1997), 443-469. [5] X. Cabré, Regularity of minimizers of semilinear elliptic problems up to dimension four, Comm. Pure Appl. Math., 63 (2010), 1362-1380.  doi: 10.1002/cpa.20327. [6] X. Cabré and A. Capella, Regularity of radial minimizers and extremal solutions of semilinear elliptic equations, Journal of Functional Analysis, 238 (2006), 709-733.  doi: 10.1016/j.jfa.2005.12.018. [7] X. Cabré, A. Figalli, X. Ros-Oton and J. Serra, Stable solutions to semilinear elliptic equations are smooth up to dimension 9, Preprint, arXiv: 1907.09403 (2019). [8] X. Cabré and X. Ros-Oton, Regularity of stable solutions up to dimension 7 in domains of double revolution, Comm. Partial Differential Equations, 38 (2013), 135-154.  doi: 10.1080/03605302.2012.697505. [9] X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians, Ⅰ: Regularity, maximum principles, and Hamiltonian estimates, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 23-53.  doi: 10.1016/j.anihpc.2013.02.001. [10] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Diff. Equ., 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306. [11] A. Capella, J. Davila, L. Dupaigne and Y. Sire, Regularity of radial extremal solutions for some non local semilinear equations, Comm. Partial Diff. Equ., 36 (2011), 1353-1384.  doi: 10.1080/03605302.2011.562954. [12] W. Chen, C. Li and B. Ou, Classification of solutions to an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343.  doi: 10.1002/cpa.20116. [13] C. Cowan, Regularity of the extremal solutions in a Gelfand system problem, Advanced Nonlinear Studies, 11 (2011), 695-700.  doi: 10.1515/ans-2011-0310. [14] C. Cowan and M. Fazly, Regularity of the extremal solutions associated to elliptic systems, Journal of Differential Equations, 257 (2014), 4087-4107.  doi: 10.1016/j.jde.2014.08.002. [15] M. G. Crandall and P. H. Rabinowitz, Some continuation and variation methods for positive solutions of nonlinear elliptic eigenvalue problems, Arch. Rat. Mech. Anal., 58 (1975), 207-218.  doi: 10.1007/BF00280741. [16] J. Davila, L. Dupaigne and J. Wei, On the fractional Lane-Emden equation, Trans. Amer. Math. Soc., 369 (2017), 6087-6104.  doi: 10.1090/tran/6872. [17] L. Dupaigne, A. Farina and B. Sirakov, Regularity of the extremal solutions for the Liouville system, Geometric Partial Differential Equations, CRM Series, Ed. Norm., Pisa., 15 (2013), 139–144. doi: 10.1007/978-88-7642-473-1_7. [18] P. Esposito, N. Ghoussoub and Y. Guo, Compactness along the branch of semi-stable and unstable solutions for an elliptic problem with a singular nonlinearity, Comm. Pure Appl. Math., 60 (2007), 1731-1768.  doi: 10.1002/cpa.20189. [19] P. Esposito, N. Ghoussoub and Y. Guo, Mathematical Analysis of Partial Differential Equations Modeling Electrostatic MEMS, Research Monograph, Courant Lecture Notes, 2010. doi: 10.1090/cln/020. [20] A. Farina, On the classification of solutions of the Lane-Emden equation on unbounded domains of $\mathbb R^n$, J. Math. Pures Appl., 87 (2007), 537-561.  doi: 10.1016/j.matpur.2007.03.001. [21] A. Farina, Stable solutions of $-\Delta u = e^u$ on $\mathbb R^n$, C. R. Math. Acad. Sci. Paris, 345 (2007), 63-66.  doi: 10.1016/j.crma.2007.05.021. [22] M. Fazly, Rigidity results for stable solutions of symmetric systems, Proceedings of the American Mathematical Society, 143 (2015), 5307-5321.  doi: 10.1090/proc/12647. [23] M. Fazly and N. Ghoussoub, De Giorgi type results for elliptic systems, Calc. Var. Partial Differential Equations, 47 (2013), 809-823.  doi: 10.1007/s00526-012-0536-x. [24] M. Fazly and J. Wei, On finite Morse index solutions of higher order fractional Lane-Emden equations, American Journal of Mathematics, 139 (2017), 433-460.  doi: 10.1353/ajm.2017.0011. [25] X. Fernández-Real and X. Ros-Oton, Boundary regularity for the fractional heat equation, Rev. Real Acad. Cienc. Ser. A Math, 110 (2016), 49-64.  doi: 10.1007/s13398-015-0218-6. [26] N. Ghoussoub and Y. Guo, On the partial differential equations of electro MEMS devices: stationary case, SIAM J. Math. Anal., 38 (2007), 1423-1449.  doi: 10.1137/050647803. [27] P. Glowacki and W. Hebisch, Pointwise estimates for densities of stable semigroups of measures, Studia Math., 104 (1992), 243-258.  doi: 10.4064/sm-104-3-243-258. [28] D. D. Joseph and T. S. Lundgren, Quasilinear Dirichlet problems driven by positive sources, Arch. Rational Mech. Anal., 49 (1973), 241-269.  doi: 10.1007/BF00250508. [29] Ph. Laurencot and C. Walker, Some singular equations modeling MEMS, Bulletin of the American Mathematical Society, 54 (2017), 437–479. doi: 10.1090/bull/1563. [30] Y. Y. Li, Remark on some conformally invariant integral equations: The method of moving spheres, J. Eur. Math. Soc. (JEMS), 6 (2004), 153-180. [31] J. Liouville, Sur l'équation aux différences partielles $\frac{d^2\log \lambda}{du dv}\pm \frac{\lambda}{2a^2}$ = 0, J. Math. Pures Appl., 18 (1853), 71-72. [32] F. Mignot and J. P. Puel, Solution radiale singuliére $-\Delta u = e^u$, C. R. Acad. Sci. Paris, 307 (1988), 379-382. [33] M. Montenegro, Minimal solutions for a class of elliptic systems, Bull. London Math. Soc., 37 (2005), 405-416.  doi: 10.1112/S0024609305004248. [34] K. Nagasaki and T. Suzuki, Spectral and related properties about the Emden-Fowler equation $-\Delta u = \lambda e^u$ on circular domains, Mathematische Annalen, 299 (1994), 1-15.  doi: 10.1007/BF01459770. [35] G. Nedev, Regularity of the extremal solution of semilinear elliptic equations, C. R. Acad. Sci. Paris. I Math., 330 (2000), 997-1002.  doi: 10.1016/S0764-4442(00)00289-5. [36] X. Ros-Oton and J. Serra, The extremal solution for the fractional Laplacian, Calc. Var. Partial Differential Equations, 50 (2014), 723-750.  doi: 10.1007/s00526-013-0653-1. [37] X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary, J. Math. Pures Appl., 101 (2014), 275-302.  doi: 10.1016/j.matpur.2013.06.003. [38] X. Ros-Oton and J. Serra, Regularity theory for general stable operators, Journal Differential Equations, 260 (2016), 8675-8715.  doi: 10.1016/j.jde.2016.02.033. [39] X. Ros-Oton, Regularity for the fractional Gelfand problem up to dimension 7, J. Math. Anal. Appl., 419 (2014), 10-19.  doi: 10.1016/j.jmaa.2014.04.048. [40] T. Sanz-Perela, Regularity of radial stable solutions to semilinear elliptic equations for the fractional Laplacian, Commun. Pure Appl. Anal., 17 (2018), 2547-2575.  doi: 10.3934/cpaa.2018121. [41] L. Silvestre, Hölder estimates for solutions of integro-differential equations like the fractional Laplace, Indiana University Mathematics Journal, 55 (2006), 1155-1174.  doi: 10.1512/iumj.2006.55.2706. [42] E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, vol. 30. Princeton University Press, Princeton, 1970. [43] S. Villegas, Boundedness of extremal solutions in dimension 4, Advances Math., 235 (2013), 126-133.  doi: 10.1016/j.aim.2012.11.015.
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