Qualitative properties of positive solutions for mixed integro-differential equations

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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

$\begin{equation}\left\{ \begin{array}{l} (-\Delta)_x^{α} u+(-\Delta)_y u+u = f(u) \ \ \ \ {\rm in}\ \ {\mathbb{R}}^N×{\mathbb{R}}^M,\\ u>0\ \ {\rm{in}}\ {\mathbb{R}}^N\times{\mathbb{R}}^M,\ \ \ \lim_{|(x,y)|\to+\infty}u(x,y) = 0, \end{array} \right.\;\;\;\left( {0.1} \right)\end{equation}$

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.

Keywords: Integro-differential equation, Hopf's Lemma, decay, symmetry.

Mathematics Subject Classification: 35R11, 35B06, 35B40, 35B50.

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.
[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.
[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.
[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.
[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.
[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.
[7]	F. Brock, Continuous steiner-symmetrization, Mathematische Nachri- chten, 172 (1995), 25-48. doi: 10.1002/mana.19951720104.
[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.
[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.
[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.
[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.
[12]	A. Ciomaga, On the strong maximum principle for second order nonlinear parabolic integro-differential equations, Adv. Diff. Eq., 17 (2012), 635-671.
[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.
[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.
[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.
[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.
[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).
[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.
[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.
[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.
[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.
[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.
[23]	N. S. Landkof, Foundations of Modern Potential Theory, Springer-Verlag, Berlin, 1972.
[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.
[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.
[26]	Y. Y. Li, Remark on some conformally invariant integral equations: The method fo moving spheres, J. Eur. Math. Soc., 6 (2004), 153-180.
[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.
[28]	H. Pham, Optimal stopping of controlled jump diffusion processes: A viscosity solution approach, J. Math. Systems Estim. Control, 8 (1998), 1-27.
[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.
[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.
[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.
[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.
[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.

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.
[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.
[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.
[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.
[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.
[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.
[7]	F. Brock, Continuous steiner-symmetrization, Mathematische Nachri- chten, 172 (1995), 25-48. doi: 10.1002/mana.19951720104.
[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.
[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.
[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.
[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.
[12]	A. Ciomaga, On the strong maximum principle for second order nonlinear parabolic integro-differential equations, Adv. Diff. Eq., 17 (2012), 635-671.
[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.
[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.
[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.
[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.
[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).
[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.
[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.
[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.
[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.
[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.
[23]	N. S. Landkof, Foundations of Modern Potential Theory, Springer-Verlag, Berlin, 1972.
[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.
[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.
[26]	Y. Y. Li, Remark on some conformally invariant integral equations: The method fo moving spheres, J. Eur. Math. Soc., 6 (2004), 153-180.
[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.
[28]	H. Pham, Optimal stopping of controlled jump diffusion processes: A viscosity solution approach, J. Math. Systems Estim. Control, 8 (1998), 1-27.
[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.
[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.
[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.
[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.
[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.

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