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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 & Continuous Dynamical Systems - A, 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.  Google Scholar

[2] J. Bertoin, Lévy Processes, Cambridge University Press, Cambridge, 1996.   Google Scholar
[3]

H. BrezisT. CazenaveY. Martel and A. Ramiandrisoa, Blow up for $u_t - \Delta u = g(u)$ revisited, Advances in Differential Equations, 1 (1996), 73-90.   Google Scholar

[4]

H. Brezis and L. Vazquez, Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Complut. Madrid, 10 (1997), 443-469.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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). Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

A. CapellaJ. DavilaL. 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.  Google Scholar

[12]

W. ChenC. Li and B. Ou, Classification of solutions to an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343.  doi: 10.1002/cpa.20116.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

J. DavilaL. Dupaigne and J. Wei, On the fractional Lane-Emden equation, Trans. Amer. Math. Soc., 369 (2017), 6087-6104.  doi: 10.1090/tran/6872.  Google Scholar

[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.  Google Scholar

[18]

P. EspositoN. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[32]

F. Mignot and J. P. Puel, Solution radiale singuliére $-\Delta u = e^u$, C. R. Acad. Sci. Paris, 307 (1988), 379-382.   Google Scholar

[33]

M. Montenegro, Minimal solutions for a class of elliptic systems, Bull. London Math. Soc., 37 (2005), 405-416.  doi: 10.1112/S0024609305004248.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[42]

E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, vol. 30. Princeton University Press, Princeton, 1970.  Google Scholar

[43]

S. Villegas, Boundedness of extremal solutions in dimension 4, Advances Math., 235 (2013), 126-133.  doi: 10.1016/j.aim.2012.11.015.  Google Scholar

show all references

References:
[1]

R. Bass, Diffusions and Elliptic Operators, Probability and its Applications, Springer-Verlag, New York, 1998.  Google Scholar

[2] J. Bertoin, Lévy Processes, Cambridge University Press, Cambridge, 1996.   Google Scholar
[3]

H. BrezisT. CazenaveY. Martel and A. Ramiandrisoa, Blow up for $u_t - \Delta u = g(u)$ revisited, Advances in Differential Equations, 1 (1996), 73-90.   Google Scholar

[4]

H. Brezis and L. Vazquez, Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Complut. Madrid, 10 (1997), 443-469.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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). Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

A. CapellaJ. DavilaL. 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.  Google Scholar

[12]

W. ChenC. Li and B. Ou, Classification of solutions to an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343.  doi: 10.1002/cpa.20116.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

J. DavilaL. Dupaigne and J. Wei, On the fractional Lane-Emden equation, Trans. Amer. Math. Soc., 369 (2017), 6087-6104.  doi: 10.1090/tran/6872.  Google Scholar

[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.  Google Scholar

[18]

P. EspositoN. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[32]

F. Mignot and J. P. Puel, Solution radiale singuliére $-\Delta u = e^u$, C. R. Acad. Sci. Paris, 307 (1988), 379-382.   Google Scholar

[33]

M. Montenegro, Minimal solutions for a class of elliptic systems, Bull. London Math. Soc., 37 (2005), 405-416.  doi: 10.1112/S0024609305004248.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[42]

E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, vol. 30. Princeton University Press, Princeton, 1970.  Google Scholar

[43]

S. Villegas, Boundedness of extremal solutions in dimension 4, Advances Math., 235 (2013), 126-133.  doi: 10.1016/j.aim.2012.11.015.  Google Scholar

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