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January  2019, 39(1): 369-393. doi: 10.3934/dcds.2019015

## Qualitative properties of positive solutions for mixed integro-differential equations

 1 Departamento de Ingeniería Matemática and Centro de Modelamiento, Matemático, Universidad de Chile, Santiago, Chile 2 Department of Mathematics, Jiangxi Normal University, Nanchang, Jiangxi 330022, China

* Corresponding author: Y. Wang

Received  January 2018 Revised  July 2018 Published  October 2018

Fund Project: P. Felmer is supported by Fondecyt Grant 1110291 and BASAL-CMM projects. Y. Wang is supported by NNSF of China, No: 11661045 and by the Jiangxi Provincial Natural Science Foundation, No: 20181BAB211002.

This paper is concerned with the qualitative properties of the solutions of mixed integro-differential equation
 $$$\left\{ \begin{array}{l} (-\Delta)_x^{α} u+(-\Delta)_y u+u = f(u) \ \ \ \ {\rm in}\ \ {\mathbb{R}}^N×{\mathbb{R}}^M,\\ u>0\ \ {\rm{in}}\ {\mathbb{R}}^N\times{\mathbb{R}}^M,\ \ \ \lim_{|(x,y)|\to+\infty}u(x,y) = 0, \end{array} \right.\;\;\;\left( {0.1} \right)$$$
with
 $N≥ 1$
,
 $M≥ 1$
and
 $α∈ (0,1)$
. We study decay and symmetry properties of the solutions to this equation. Difficulties arise due to the mixed character of the integro-differential operators. Here, a crucial role is played by a version of the Hopf's Lemma we prove in our setting. In studying the decay, we construct appropriate super and sub solutions and we use the moving planes method to prove the symmetry properties.
Citation: Patricio Felmer, Ying Wang. Qualitative properties of positive solutions for mixed integro-differential equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 369-393. doi: 10.3934/dcds.2019015
##### References:
 [1] G. Barles, R. Buckdahn and E. Pardoux, Backward stochastic differential equations and integral partial differential equations, Stochastics Stochastics Rep., 60 (1997), 57-83.  doi: 10.1080/17442509708834099.  Google Scholar [2] G. Barles, E. Chasseigne, A. Ciomaga and C. Imbert, Lipschitz regularity of solutions for mixed integro-differential equations, J. Diff. Eq., 252 (2012), 6012-6060.  doi: 10.1016/j.jde.2012.02.013.  Google Scholar [3] G. Barles, E. Chasseigne, A. Ciomaga and C. Imbert, Large time behavior of periodic viscosity solutions for uniformly elliptic integro-differential equations, Calc. Var. Part. Diff. Eq., 50 (2014), 283-304.  doi: 10.1007/s00526-013-0636-2.  Google Scholar [4] F. Benth, K. Karlsen and K. Reikvam, Optimal portfolio selection with consumption and nonlinear integro-differential equations with gradient constraint: A viscosity solution approach, Finance Stoch., 5 (2001), 275-303.  doi: 10.1007/PL00013538.  Google Scholar [5] H. Berestycki and P. L. Lions, Non linear scalar field equations Ⅰ: Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345.  doi: 10.1007/BF00250555.  Google Scholar [6] H. Berestycki and L. Nirenberg, On the method of moving planes and the sliding method, Bol. Soc. Brasileira Mat., 22 (1991), 1-37.  doi: 10.1007/BF01244896.  Google Scholar [7] F. Brock, Continuous steiner-symmetrization, Mathematische Nachri- chten, 172 (1995), 25-48.  doi: 10.1002/mana.19951720104.  Google Scholar [8] J. Busca and P. Felmer, Qualitative properties of some bounded positive solutions to scalar field equations, Calc. Var. Partial Differential Equations, 13 (2001), 191-211.  doi: 10.1007/PL00009928.  Google Scholar [9] X. Cabré and Y. Sire, Nonlinear equations for fractional laplacians Ⅱ: Existence, uniqueness and qualitative properties of solutions, Trans. Amer. Math. Soc., 367 (2014), 911-941.  doi: 10.1090/S0002-9947-2014-05906-0.  Google Scholar [10] L. Caffarelli and L. 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Valdinoci, Existence and symmetry results for a schrödinger type problem involving the fractional laplacian, Le Matematiche (Catania), 68 (2013), 201-206.   Google Scholar [16] J. Dolbeault and P. Felmer, Symmetry and monotonicity properties for positive solution of semi-linear elliptic PDE's, Comm. Part. Diff. Eq., 25 (2000), 1153-1169.  doi: 10.1080/03605300008821545.  Google Scholar [17] A. Esfahani and S. E. Esfahani, Positive and nodal solutions of the generalized BO-ZK equation, Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales, Serie A. Matematicas, DOI: 10.1007/s13398-017-0435-2(2017). Google Scholar [18] P. Felmer, A. Quaas and J. Tan, Positive solutions of non-linear Schrödinger equation with the fractional laplacian, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 142 (2012), 1237-1262.  doi: 10.1017/S0308210511000746.  Google Scholar [19] P. Felmer and Y. Wang, Radial symmetry of positive solutions to equations involving the fractional laplacian, Comm. Contem. Math., 16 (2014), 1350023, 24 pp. doi: 10.1142/S0219199713500235.  Google Scholar [20] B. Gidas, W. M. Ni and L. Nirenberg, Symmetry of positive solutions of non-linear elliptic equations in ${\mathbb{R}}^N$, Math. Anal. Appl., Part A, Advances in Math. Suppl. Studied, 7A (1981), 369-402.   Google Scholar [21] C. Gui, Symmetry of the blow up set of a porous medium type equation, Comm. Pure Appl. Math., 48 (1995), 471-500.  doi: 10.1002/cpa.3160480502.  Google Scholar [22] M. K. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p = 0$ in ${\mathbb{R}}^N$, Arch. Rational Mech. Anal., 105 (1989), 243-266.  doi: 10.1007/BF00251502.  Google Scholar [23] N. S. Landkof, Foundations of Modern Potential Theory, Springer-Verlag, Berlin, 1972.  Google Scholar [24] C. M. Li, Monotonicity and symmetry of solutions of fully non-linear elliptic equations on unbounded domains, Comm. Part. Diff. Eq., 16 (1991), 585-615.  doi: 10.1080/03605309108820770.  Google Scholar [25] Y. Li and W. M. Ni, Radial symmetry of positive solutions of non-linear elliptic equations in ${\mathbb{R}}^N$, Comm. Part. Diff. Eq., 18 (1993), 1043-1054.  doi: 10.1080/03605309308820960.  Google Scholar [26] Y. Y. Li, Remark on some conformally invariant integral equations: The method fo moving spheres, J. Eur. Math. Soc., 6 (2004), 153-180.   Google Scholar [27] F. Pacella and M. Ramaswamy, Symmetry of solutions of elliptic equations via maximum principles, Handbook of Differential Equations (M. Chipot, ed.), Elsevier, 4 (2008), 269-312.  doi: 10.1016/S1874-5733(08)80021-6.  Google Scholar [28] H. Pham, Optimal stopping of controlled jump diffusion processes: A viscosity solution approach, J. Math. Systems Estim. Control, 8 (1998), 1-27.   Google Scholar [29] A. Quaas and A. Xia, Liouville type theorems for nonlinear elliptic equations and systems involving fractional laplacian in the half space, Calc. Var. Part. Diff. Eq., 52 (2015), 641-659.  doi: 10.1007/s00526-014-0727-8.  Google Scholar [30] 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 [31] R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137.  doi: 10.3934/dcds.2013.33.2105.  Google Scholar [32] R. Servadei and E. Valdinoci, A Brezis-Nirenberg result for non-local critical equations in low dimension, Comm. Pure Appl. Anal., 12 (2013), 2445-2464.  doi: 10.3934/cpaa.2013.12.2445.  Google Scholar [33] Y. Sire and E. Valdinoci, Fractional laplacian phase transitions and boundary reactions: a geometric inequality and a symmetry result, J. Funct. Anal., 256 (2009), 1842-1864.  doi: 10.1016/j.jfa.2009.01.020.  Google Scholar

show all references

##### References:
 [1] G. Barles, R. Buckdahn and E. Pardoux, Backward stochastic differential equations and integral partial differential equations, Stochastics Stochastics Rep., 60 (1997), 57-83.  doi: 10.1080/17442509708834099.  Google Scholar [2] G. Barles, E. Chasseigne, A. Ciomaga and C. Imbert, Lipschitz regularity of solutions for mixed integro-differential equations, J. Diff. Eq., 252 (2012), 6012-6060.  doi: 10.1016/j.jde.2012.02.013.  Google Scholar [3] G. Barles, E. Chasseigne, A. Ciomaga and C. Imbert, Large time behavior of periodic viscosity solutions for uniformly elliptic integro-differential equations, Calc. Var. Part. Diff. Eq., 50 (2014), 283-304.  doi: 10.1007/s00526-013-0636-2.  Google Scholar [4] F. Benth, K. Karlsen and K. Reikvam, Optimal portfolio selection with consumption and nonlinear integro-differential equations with gradient constraint: A viscosity solution approach, Finance Stoch., 5 (2001), 275-303.  doi: 10.1007/PL00013538.  Google Scholar [5] H. Berestycki and P. L. Lions, Non linear scalar field equations Ⅰ: Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345.  doi: 10.1007/BF00250555.  Google Scholar [6] H. Berestycki and L. Nirenberg, On the method of moving planes and the sliding method, Bol. Soc. Brasileira Mat., 22 (1991), 1-37.  doi: 10.1007/BF01244896.  Google Scholar [7] F. Brock, Continuous steiner-symmetrization, Mathematische Nachri- chten, 172 (1995), 25-48.  doi: 10.1002/mana.19951720104.  Google Scholar [8] J. Busca and P. Felmer, Qualitative properties of some bounded positive solutions to scalar field equations, Calc. Var. Partial Differential Equations, 13 (2001), 191-211.  doi: 10.1007/PL00009928.  Google Scholar [9] X. Cabré and Y. Sire, Nonlinear equations for fractional laplacians Ⅱ: Existence, uniqueness and qualitative properties of solutions, Trans. Amer. Math. Soc., 367 (2014), 911-941.  doi: 10.1090/S0002-9947-2014-05906-0.  Google Scholar [10] 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.  Google Scholar [11] 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.  Google Scholar [12] A. Ciomaga, On the strong maximum principle for second order nonlinear parabolic integro-differential equations, Adv. Diff. Eq., 17 (2012), 635-671.   Google Scholar [13] C. Cortázar, M. Elgueta and P. Felmer, On a semilinear elliptic problem in ${\mathbb{R}}^N$ with a non-lipschitzian non-linearity, Adv. Diff. Eq., 1 (1996), 199-218.   Google Scholar [14] F. Da Lio and B. Sirakov, Symmetry results for viscosity solutions of fully nonlinear uniformly elliptic equations, J. Eur. Math. Soc., 9 (2007), 317-330.  doi: 10.4171/JEMS/81.  Google Scholar [15] S. Dipierro, G. Palatucci and E. Valdinoci, Existence and symmetry results for a schrödinger type problem involving the fractional laplacian, Le Matematiche (Catania), 68 (2013), 201-206.   Google Scholar [16] J. Dolbeault and P. Felmer, Symmetry and monotonicity properties for positive solution of semi-linear elliptic PDE's, Comm. Part. Diff. Eq., 25 (2000), 1153-1169.  doi: 10.1080/03605300008821545.  Google Scholar [17] A. Esfahani and S. E. Esfahani, Positive and nodal solutions of the generalized BO-ZK equation, Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales, Serie A. Matematicas, DOI: 10.1007/s13398-017-0435-2(2017). Google Scholar [18] P. Felmer, A. Quaas and J. Tan, Positive solutions of non-linear Schrödinger equation with the fractional laplacian, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 142 (2012), 1237-1262.  doi: 10.1017/S0308210511000746.  Google Scholar [19] P. Felmer and Y. Wang, Radial symmetry of positive solutions to equations involving the fractional laplacian, Comm. Contem. Math., 16 (2014), 1350023, 24 pp. doi: 10.1142/S0219199713500235.  Google Scholar [20] B. Gidas, W. M. Ni and L. Nirenberg, Symmetry of positive solutions of non-linear elliptic equations in ${\mathbb{R}}^N$, Math. Anal. Appl., Part A, Advances in Math. Suppl. Studied, 7A (1981), 369-402.   Google Scholar [21] C. Gui, Symmetry of the blow up set of a porous medium type equation, Comm. Pure Appl. Math., 48 (1995), 471-500.  doi: 10.1002/cpa.3160480502.  Google Scholar [22] M. K. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p = 0$ in ${\mathbb{R}}^N$, Arch. Rational Mech. Anal., 105 (1989), 243-266.  doi: 10.1007/BF00251502.  Google Scholar [23] N. S. Landkof, Foundations of Modern Potential Theory, Springer-Verlag, Berlin, 1972.  Google Scholar [24] C. M. Li, Monotonicity and symmetry of solutions of fully non-linear elliptic equations on unbounded domains, Comm. Part. Diff. Eq., 16 (1991), 585-615.  doi: 10.1080/03605309108820770.  Google Scholar [25] Y. Li and W. M. Ni, Radial symmetry of positive solutions of non-linear elliptic equations in ${\mathbb{R}}^N$, Comm. Part. Diff. Eq., 18 (1993), 1043-1054.  doi: 10.1080/03605309308820960.  Google Scholar [26] Y. Y. Li, Remark on some conformally invariant integral equations: The method fo moving spheres, J. Eur. Math. Soc., 6 (2004), 153-180.   Google Scholar [27] F. Pacella and M. Ramaswamy, Symmetry of solutions of elliptic equations via maximum principles, Handbook of Differential Equations (M. Chipot, ed.), Elsevier, 4 (2008), 269-312.  doi: 10.1016/S1874-5733(08)80021-6.  Google Scholar [28] H. Pham, Optimal stopping of controlled jump diffusion processes: A viscosity solution approach, J. Math. Systems Estim. Control, 8 (1998), 1-27.   Google Scholar [29] A. Quaas and A. Xia, Liouville type theorems for nonlinear elliptic equations and systems involving fractional laplacian in the half space, Calc. Var. Part. Diff. Eq., 52 (2015), 641-659.  doi: 10.1007/s00526-014-0727-8.  Google Scholar [30] 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 [31] R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137.  doi: 10.3934/dcds.2013.33.2105.  Google Scholar [32] R. Servadei and E. Valdinoci, A Brezis-Nirenberg result for non-local critical equations in low dimension, Comm. Pure Appl. Anal., 12 (2013), 2445-2464.  doi: 10.3934/cpaa.2013.12.2445.  Google Scholar [33] Y. Sire and E. Valdinoci, Fractional laplacian phase transitions and boundary reactions: a geometric inequality and a symmetry result, J. Funct. Anal., 256 (2009), 1842-1864.  doi: 10.1016/j.jfa.2009.01.020.  Google Scholar
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