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# Regularity of extremal solutions of nonlocal elliptic systems

• 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.

Mathematics Subject Classification: 35R09, 35R11, 35B45, 35B65, 35J50.

 Citation:

•  [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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