The Nehari manifold for fractional systems involving critical nonlinearities

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July 2016, 15(4): 1285-1308. doi: 10.3934/cpaa.2016.15.1285

The Nehari manifold for fractional systems involving critical nonlinearities

Xiaoming He ^1, , Marco Squassina ^2, and Wenming Zou ^3,

1.	College of Science, Minzu University of China, Beijing 100081, China
2.	Dipartimento di Informatica, Università degli Studi di Verona, Cá Vignal 2, Strada Le Grazie 15, I-37134 Veron
3.	Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China

Received September 2015 Revised January 2016 Published April 2016

Abstract
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We study the combined effect of concave and convex nonlinearities on the number of positive solutions for a fractional system involving critical Sobolev exponents. With the help of the Nehari manifold, we prove that the system admits at least two positive solutions when the pair of parameters $(\lambda,\mu)$ belongs to a suitable subset of $R^2$.

Keywords: Nehari manifold., concave-convex nonlinearities, Fractional systems.

Mathematics Subject Classification: 47G20, 35J50, 35B6.

Citation: Xiaoming He, Marco Squassina, Wenming Zou. The Nehari manifold for fractional systems involving critical nonlinearities. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1285-1308. doi: 10.3934/cpaa.2016.15.1285

References:

[1]	C.O. Alves, D.C. de Morais Filho and M.A.S. Souto, On systems of elliptic equations involving subcritical or critical Sobolev exponents,, \emph{Nonlinear Anal.}, 42 (2000), 771. doi: 10.1016/S0362-546X(99)00121-2.
[2]	A. Ambrosetti, H. Brezis and G. Cerami, Combined effects of concave-convex nonlinearities in some elliptic problems,, \emph{J. Funct. Anal.}, 122 (1994), 519. doi: 10.1006/jfan.1994.1078.
[3]	B. Barrios, E. Colorado, A. de Pablo and U. Sánchez, On some critical problems for the fractional Laplacian,, \emph{J. Differential Equations}, 252 (2012), 6133. doi: 10.1016/j.jde.2012.02.023.
[4]	B. Barrios, E. Colorado, R. Servadei and F. Soria, A critical fractional equation with concave-convex power nonlinearities,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, 32 (2015), 875. doi: 10.1016/j.anihpc.2014.04.003.
[5]	C. Brändle, E. Colorado and A. de Pablo, A concave-convex elliptic problem involving the fractional Laplacian,, \emph{Proc. Roy. Soc. Edinburgh Sect. A. Math.}, 142 (2013), 39.
[6]	H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents,, \emph{Comm. Pure Appl. Math.}, 36 (1983), 437. doi: 10.1002/cpa.3160360405.
[7]	K.J. Brown and Y. Zhang, The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function,, \emph{J. Differential Equations}, 193 (2003), 481. doi: 10.1016/S0022-0396(03)00121-9.
[8]	X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian,, \emph{Adv. Math.}, 224 (2010), 2052. doi: 10.1016/j.aim.2010.01.025.
[9]	L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian,, \emph{Comm. Partial Differential Equations}, 32 (2007), 1245. doi: 10.1080/03605300600987306.
[10]	A. Capella, J. Dávila, L. Dupaigne and Y. Sire, Regularity of radial extremal solutions for some non-local semilinear equations,, \emph{Comm. Partial Differential Equations}, 36 (2011), 1353. doi: 10.1080/03605302.2011.562954.
[11]	W. Chen and S. Deng, The Nehari manifold for a nonlinear elliptic operators involving concave-convex nonlinearities,, \emph{Z. Angew. Math. Phys.}, 66 (2015), 1387. doi: 10.1007/s00033-014-0486-6.
[12]	W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation,, \emph{Comm. Pure Appl. Math.}, 59 (2006), 330. doi: 10.1002/cpa.20116.
[13]	X. Cheng and S. Ma, Existence of three nontrivial solutions for elliptic systems with critcal exponents and weights,, \emph{Nonlinear Anal.}, 69 (2008), 3537. doi: 10.1016/j.na.2007.09.040.
[14]	E. Colorado, A. de Pablo and U. Sánchez, Perturbation of a critical fractional equations,, \emph{Pacific J. Math.}, 271 (2014), 65. doi: 10.2140/pjm.2014.271.65.
[15]	A. Cotsiolis and N. Tavoularis, Best constant for Sobolev inequalities for higher order fractional derivatives,, \emph{J. Math. Anal. Appl.}, 295 (2004), 225. doi: 10.1016/j.jmaa.2004.03.034.
[16]	P. Drabek and S.I. Pohozaev, Positive solutions for the $p$-Laplacian: application of the fibering methods,, \emph{Proc. Roy. Soc. Edinburgh Sect. A}, 127 (1997), 703. doi: 10.1017/S0308210500023787.
[17]	I. Ekeland, On the variational principle,, \emph{J. Math. Anal. Appl.}, 47 (1974), 324.
[18]	L. Faria, O. Miyagaki, F. Pereira, M. Squassina and C. Zhang, The Brezis-Nirenberg problem for nonlocal systems,, \emph{Adv. Nonlinear Anal.}, 5 (2016), 85. doi: 10.1515/anona-2015-0114.
[19]	P. Han, The effect of the domain topology of the number of positive solutions of elliptic systems involving critical Sobolev exponents,, \emph{Houston J. Math.}, 32 (2006), 1241.
[20]	T. Hsu and H. Lin, Multiple positive solutions for a critical elliptic system with concave-convex nonlinearities,, \emph{Proc. Roy. Soc. Edinburgh Sect. A}, 139 (2009), 1163. doi: 10.1017/S0308210508000875.
[21]	E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces,, \emph{Bull. Sci. Math.}, 136 (2012), 521. doi: 10.1016/j.bulsci.2011.12.004.
[22]	X. Shang, J. Zhang and Y. Yang, Positive solutions of nonhomogeneous fractional Laplacian problem with critical exponent,, \emph{Comm. Pure Appl. Anal.}, 13 (2014), 567. doi: 10.3934/cpaa.2014.13.567.
[23]	R. Servadei and E. Valdinoci, Mountain pass solutions for nonlinear elliptic operators,, \emph{J. Math. Anal. Appl.}, 389 (2012), 887. doi: 10.1016/j.jmaa.2011.12.032.
[24]	R. Servadei and E. Valdinoci, The Brezis-Nirenberg result for the fractional Laplacian,, \emph{Trans. Amer. Math. Soc.}, 367 (2015), 67. doi: 10.1090/S0002-9947-2014-05884-4.
[25]	R. Servadei and E. Valdinoci, On the spectrum of two different fractional operators,, \emph{Proc. Roy. Soc. Edinburgh Sect. A}, 144 (2014), 831. doi: 10.1017/S0308210512001783.
[26]	J. Serra and X. Ros-Oton, The Dirichlet problem for the fractional Laplacian: regularity up to the boundary,, \emph{J. Math. Pures Appl.}, 101 (2014), 275. doi: 10.1016/j.matpur.2013.06.003.
[27]	L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator,, \emph{Comm. Pure Appl. Math.}, 60 (2007), 67. doi: 10.1002/cpa.20153.
[28]	J. Tan, The Brézis-Nirenberg type problem involving the square root of the Laplacian,, \emph{Calc. Var. Partial Differential Equations}, 36 (2011), 21. doi: 10.1007/s00526-010-0378-3.
[29]	Y. Wei and X. Su, Multiplicity of solutions for non-local elliptic equations driven by the fractional Laplacian,, \emph{Calc. Var. Partial Differential Equations}, 52 (2015), 95. doi: 10.1007/s00526-013-0706-5.
[30]	T.F. Wu, On semilinear elliptic equations involving concave-convex nonlinearities and sign-changing weight function,, \emph{J. Math. Anal. Appl.}, 318 (2006), 253. doi: 10.1016/j.jmaa.2005.05.057.
[31]	T.F. Wu, The Nehari manifold for a semilinear elliptic system involving sign-changing weight functions,, \emph{Nonlinear Anal.}, 68 (2008), 1733. doi: 10.1016/j.na.2007.01.004.
[32]	X. Yu, The Nehari manifold for elliptic equation involving the square root of the laplacian,, \emph{J. Differential Equations}, 252 (2012), 1283. doi: 10.1016/j.jde.2011.09.015.

show all references

References:

[1]	C.O. Alves, D.C. de Morais Filho and M.A.S. Souto, On systems of elliptic equations involving subcritical or critical Sobolev exponents,, \emph{Nonlinear Anal.}, 42 (2000), 771. doi: 10.1016/S0362-546X(99)00121-2.
[2]	A. Ambrosetti, H. Brezis and G. Cerami, Combined effects of concave-convex nonlinearities in some elliptic problems,, \emph{J. Funct. Anal.}, 122 (1994), 519. doi: 10.1006/jfan.1994.1078.
[3]	B. Barrios, E. Colorado, A. de Pablo and U. Sánchez, On some critical problems for the fractional Laplacian,, \emph{J. Differential Equations}, 252 (2012), 6133. doi: 10.1016/j.jde.2012.02.023.
[4]	B. Barrios, E. Colorado, R. Servadei and F. Soria, A critical fractional equation with concave-convex power nonlinearities,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, 32 (2015), 875. doi: 10.1016/j.anihpc.2014.04.003.
[5]	C. Brändle, E. Colorado and A. de Pablo, A concave-convex elliptic problem involving the fractional Laplacian,, \emph{Proc. Roy. Soc. Edinburgh Sect. A. Math.}, 142 (2013), 39.
[6]	H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents,, \emph{Comm. Pure Appl. Math.}, 36 (1983), 437. doi: 10.1002/cpa.3160360405.
[7]	K.J. Brown and Y. Zhang, The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function,, \emph{J. Differential Equations}, 193 (2003), 481. doi: 10.1016/S0022-0396(03)00121-9.
[8]	X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian,, \emph{Adv. Math.}, 224 (2010), 2052. doi: 10.1016/j.aim.2010.01.025.
[9]	L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian,, \emph{Comm. Partial Differential Equations}, 32 (2007), 1245. doi: 10.1080/03605300600987306.
[10]	A. Capella, J. Dávila, L. Dupaigne and Y. Sire, Regularity of radial extremal solutions for some non-local semilinear equations,, \emph{Comm. Partial Differential Equations}, 36 (2011), 1353. doi: 10.1080/03605302.2011.562954.
[11]	W. Chen and S. Deng, The Nehari manifold for a nonlinear elliptic operators involving concave-convex nonlinearities,, \emph{Z. Angew. Math. Phys.}, 66 (2015), 1387. doi: 10.1007/s00033-014-0486-6.
[12]	W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation,, \emph{Comm. Pure Appl. Math.}, 59 (2006), 330. doi: 10.1002/cpa.20116.
[13]	X. Cheng and S. Ma, Existence of three nontrivial solutions for elliptic systems with critcal exponents and weights,, \emph{Nonlinear Anal.}, 69 (2008), 3537. doi: 10.1016/j.na.2007.09.040.
[14]	E. Colorado, A. de Pablo and U. Sánchez, Perturbation of a critical fractional equations,, \emph{Pacific J. Math.}, 271 (2014), 65. doi: 10.2140/pjm.2014.271.65.
[15]	A. Cotsiolis and N. Tavoularis, Best constant for Sobolev inequalities for higher order fractional derivatives,, \emph{J. Math. Anal. Appl.}, 295 (2004), 225. doi: 10.1016/j.jmaa.2004.03.034.
[16]	P. Drabek and S.I. Pohozaev, Positive solutions for the $p$-Laplacian: application of the fibering methods,, \emph{Proc. Roy. Soc. Edinburgh Sect. A}, 127 (1997), 703. doi: 10.1017/S0308210500023787.
[17]	I. Ekeland, On the variational principle,, \emph{J. Math. Anal. Appl.}, 47 (1974), 324.
[18]	L. Faria, O. Miyagaki, F. Pereira, M. Squassina and C. Zhang, The Brezis-Nirenberg problem for nonlocal systems,, \emph{Adv. Nonlinear Anal.}, 5 (2016), 85. doi: 10.1515/anona-2015-0114.
[19]	P. Han, The effect of the domain topology of the number of positive solutions of elliptic systems involving critical Sobolev exponents,, \emph{Houston J. Math.}, 32 (2006), 1241.
[20]	T. Hsu and H. Lin, Multiple positive solutions for a critical elliptic system with concave-convex nonlinearities,, \emph{Proc. Roy. Soc. Edinburgh Sect. A}, 139 (2009), 1163. doi: 10.1017/S0308210508000875.
[21]	E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces,, \emph{Bull. Sci. Math.}, 136 (2012), 521. doi: 10.1016/j.bulsci.2011.12.004.
[22]	X. Shang, J. Zhang and Y. Yang, Positive solutions of nonhomogeneous fractional Laplacian problem with critical exponent,, \emph{Comm. Pure Appl. Anal.}, 13 (2014), 567. doi: 10.3934/cpaa.2014.13.567.
[23]	R. Servadei and E. Valdinoci, Mountain pass solutions for nonlinear elliptic operators,, \emph{J. Math. Anal. Appl.}, 389 (2012), 887. doi: 10.1016/j.jmaa.2011.12.032.
[24]	R. Servadei and E. Valdinoci, The Brezis-Nirenberg result for the fractional Laplacian,, \emph{Trans. Amer. Math. Soc.}, 367 (2015), 67. doi: 10.1090/S0002-9947-2014-05884-4.
[25]	R. Servadei and E. Valdinoci, On the spectrum of two different fractional operators,, \emph{Proc. Roy. Soc. Edinburgh Sect. A}, 144 (2014), 831. doi: 10.1017/S0308210512001783.
[26]	J. Serra and X. Ros-Oton, The Dirichlet problem for the fractional Laplacian: regularity up to the boundary,, \emph{J. Math. Pures Appl.}, 101 (2014), 275. doi: 10.1016/j.matpur.2013.06.003.
[27]	L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator,, \emph{Comm. Pure Appl. Math.}, 60 (2007), 67. doi: 10.1002/cpa.20153.
[28]	J. Tan, The Brézis-Nirenberg type problem involving the square root of the Laplacian,, \emph{Calc. Var. Partial Differential Equations}, 36 (2011), 21. doi: 10.1007/s00526-010-0378-3.
[29]	Y. Wei and X. Su, Multiplicity of solutions for non-local elliptic equations driven by the fractional Laplacian,, \emph{Calc. Var. Partial Differential Equations}, 52 (2015), 95. doi: 10.1007/s00526-013-0706-5.
[30]	T.F. Wu, On semilinear elliptic equations involving concave-convex nonlinearities and sign-changing weight function,, \emph{J. Math. Anal. Appl.}, 318 (2006), 253. doi: 10.1016/j.jmaa.2005.05.057.
[31]	T.F. Wu, The Nehari manifold for a semilinear elliptic system involving sign-changing weight functions,, \emph{Nonlinear Anal.}, 68 (2008), 1733. doi: 10.1016/j.na.2007.01.004.
[32]	X. Yu, The Nehari manifold for elliptic equation involving the square root of the laplacian,, \emph{J. Differential Equations}, 252 (2012), 1283. doi: 10.1016/j.jde.2011.09.015.

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