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

The Nehari manifold for fractional systems involving critical nonlinearities

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

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 $R^2$.
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

I. Ekeland, On the variational principle,, \emph{J. Math. Anal. Appl.}, 47 (1974), 324.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

I. Ekeland, On the variational principle,, \emph{J. Math. Anal. Appl.}, 47 (1974), 324.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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