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June  2018, 11(3): 391-424. doi: 10.3934/dcdss.2018022

Global compactness results for nonlocal problems

Lorenzo Brasco 1,2, , Marco Squassina 3, and  Yang Yang 4,

1. 

Dipartimento di Matematica e Informatica, Università degli Studi di Ferrara, Via Machiavelli 35,44121 Ferrara, Italy
2. 

Aix-Marseille Université, CNRS, Centrale Marseille, I2M, UMR 7373, 39 Rue Frédéric Joliot Curie, 13453 Marseille, France
3. 

Dipartimento di Matematica e Fisica, Università Cattolica del Sacro Cuore, Via dei Musei 41, I-25121 Brescia, Italy
4. 

School of Science, Jiangnan University, Wuxi, Jiangsu 214122, China

Received  May 2017 Revised  August 2017 Published  October 2017

Fund Project: L.B. and M.S. are members of the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). Y.Y. was supported by NSFC (No. 11501252,11571176), Tian Yuan Special Foundation (No. 11226116), Natural Science Foundation of Jiangsu Province of China for Young Scholars (No. BK2012109).

We obtain a Struwe type global compactness result for a class of nonlinear nonlocal problems involving the fractional $p-$Laplacian operator and nonlinearities at critical growth.

Keywords: Fractional $p$-Laplacian, variational methods, global compactness.
Mathematics Subject Classification: Primary: 35R11, 35J62, 35B33; Secondary: 35A15.
Citation: Lorenzo Brasco, Marco Squassina, Yang Yang. Global compactness results for nonlocal problems. Discrete and Continuous Dynamical Systems - S, 2018, 11 (3) : 391-424. doi: 10.3934/dcdss.2018022
References:
[1]

C. O. Alves, Existence of positive solutions for a problem with lack of compactness involving the $p$-Laplacian, Nonlinear Anal., 51 (2002), 1187-1206.  doi: 10.1016/S0362-546X(01)00887-2.

[2]

L. Brasco and G. Franzina, Convexity properties of Dirichlet integrals and Picone-type inequalities, Kodai Math. J., 37 (2014), 769-799.  doi: 10.2996/kmj/1414674621.

[3]

L. Brasco, E. Lindgren and E. Parini, The fractional Cheeger problem, Interfaces Free Bound., 16 (2014), 419-458.  doi: 10.4171/IFB/325.

[4]

L. Brasco, S. Mosconi and M. Squassina, Optimal decay of extremals for the fractional Sobolev inequality Calc. Var. Partial Differential Equations 55 (2016), Art. 23, 32 pp. doi: 10.1007/s00526-016-0958-y.

[5]

L. Brasco and E. Parini, The second eigenvalue of the fractional $p$-Laplacian, Adv. Calc. Var., 9 (2016), 323-355.  doi: 10.1515/acv-2015-0007.

[6]

H. Brézis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490.  doi: 10.2307/2044999.

[7]

M. Clapp, A global compactness result for elliptic problems with critical nonlinearity on symmetric domains, Nonlinear Equations: Methods, Models and Applications (Bergamo, 2001), 117-126, Progr. Nonlinear Differential Equations Appl. , 54, Birkhäuser, Basel, 2003.

[8]

A. Di Castro, T. Kuusi and G. Palatucci, Local behavior of fractional $p$-minimizers, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 1279-1299.  doi: 10.1016/j.anihpc.2015.04.003.

[9]

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.

[10]

M. M. Fall and T. Weth, Nonexistence results for a class of fractional elliptic boundary value problems, J. Funct. Anal., 263 (2012), 2205-2227.  doi: 10.1016/j.jfa.2012.06.018.

[11]

R. L. Frank and R. Seiringer, Non-linear ground state representations and sharp Hardy inequalities, J. Funct. Anal., 255 (2008), 3407-3430.  doi: 10.1016/j.jfa.2008.05.015.

[12]

F. Gazzola, H. C. Grunau and M. Squassina, Existence and nonexistence results for critical growth biharmonic elliptic equations, Calc. Var. Partial Differential Equations, 18 (2003), 117-143.  doi: 10.1007/s00526-002-0182-9.

[13]

P. Gerard, Description du défaut de compacité de l'injection de Sobolev, ESAIM Control Optim. Calc. Var., 3 (1998), 213-233.  doi: 10.1051/cocv:1998107.

[14]

S. Jaffard, Analysis of the lack of compactness in the critical Sobolev embeddings, J. Funct. Anal., 161 (1999), 384-396.  doi: 10.1006/jfan.1998.3364.

[15]

C. Mercuri, B. Sciunzi and M. Squassina, On Coron's problem for the $p$-Laplacian, J. Math. Anal. Appl., 421 (2015), 362-369.  doi: 10.1016/j.jmaa.2014.07.018.

[16]

C. Mercuri and M. Willem, A global compactness result for the $p$-Laplacian involving critical nonlinearities, Discrete Cont. Dyn. Syst., 28 (2010), 469-493.  doi: 10.3934/dcds.2010.28.469.

[17]

G. Palatucci and A. Pisante, Improved Sobolev embeddings, profile decomposition, and concentration compactness for fractional Sobolev spaces, Calc. Var. Partial Differential Equations, 50 (2014), 799-829.  doi: 10.1007/s00526-013-0656-y.

[18]

G. Palatucci and A. Pisante, A global compactness type result for Palais-Smale sequences in fractional Sobolev spaces, Nonlinear Anal., 117 (2015), 1-7.  doi: 10.1016/j.na.2014.12.027.

[19]

S. Secchi, N. Shioji and M. Squassina, Coron problem for fractional equations, Differential Integral Equations, 28 (2015), 103-118. 

[20]

W. Sickel, L. Skrzypczak and J. Vybiral, On the interplay of regularity and decay in case of radial functions Ⅰ. Inhomogeneous spaces Commun. Contemp. Math. 14 (2012), 1250005, 60 pp. doi: 10.1142/S0219199712500058.

[21]

M. Struwe, A global compactness result for elliptic boundary value problems involving limiting nonlinearities, Math. Z., 187 (1984), 511-517.  doi: 10.1007/BF01174186.

[22]

H. Triebel, Theory of Function Spaces. III, Monographs in Mathematics, 100. Birkhäuser Verlag, Basel, 2006.

[23]

H. Triebel, Theory of Function Spaces [Reprint of 1983 edition]. Modern Birkhäuser Classics. Birkhäuser/Springer Basel AG, Basel, 2010.

[24]

M. Willem, Minimax Theorems, Progress Nonlinear Differential Equations Appl. 24. Birkhäuser Boston, Inc., Boston, MA, 1996. doi: 10.1007/978-1-4612-4146-1.

[25]

S. Yan, A global compactness result for quasilinear elliptic equations with critical Sobolev exponents, Chinese Ann. Math. Ser. A, 16 (1995), 397-402. 

show all references

References:
[1]

C. O. Alves, Existence of positive solutions for a problem with lack of compactness involving the $p$-Laplacian, Nonlinear Anal., 51 (2002), 1187-1206.  doi: 10.1016/S0362-546X(01)00887-2.

[2]

L. Brasco and G. Franzina, Convexity properties of Dirichlet integrals and Picone-type inequalities, Kodai Math. J., 37 (2014), 769-799.  doi: 10.2996/kmj/1414674621.

[3]

L. Brasco, E. Lindgren and E. Parini, The fractional Cheeger problem, Interfaces Free Bound., 16 (2014), 419-458.  doi: 10.4171/IFB/325.

[4]

L. Brasco, S. Mosconi and M. Squassina, Optimal decay of extremals for the fractional Sobolev inequality Calc. Var. Partial Differential Equations 55 (2016), Art. 23, 32 pp. doi: 10.1007/s00526-016-0958-y.

[5]

L. Brasco and E. Parini, The second eigenvalue of the fractional $p$-Laplacian, Adv. Calc. Var., 9 (2016), 323-355.  doi: 10.1515/acv-2015-0007.

[6]

H. Brézis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490.  doi: 10.2307/2044999.

[7]

M. Clapp, A global compactness result for elliptic problems with critical nonlinearity on symmetric domains, Nonlinear Equations: Methods, Models and Applications (Bergamo, 2001), 117-126, Progr. Nonlinear Differential Equations Appl. , 54, Birkhäuser, Basel, 2003.

[8]

A. Di Castro, T. Kuusi and G. Palatucci, Local behavior of fractional $p$-minimizers, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 1279-1299.  doi: 10.1016/j.anihpc.2015.04.003.

[9]

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.

[10]

M. M. Fall and T. Weth, Nonexistence results for a class of fractional elliptic boundary value problems, J. Funct. Anal., 263 (2012), 2205-2227.  doi: 10.1016/j.jfa.2012.06.018.

[11]

R. L. Frank and R. Seiringer, Non-linear ground state representations and sharp Hardy inequalities, J. Funct. Anal., 255 (2008), 3407-3430.  doi: 10.1016/j.jfa.2008.05.015.

[12]

F. Gazzola, H. C. Grunau and M. Squassina, Existence and nonexistence results for critical growth biharmonic elliptic equations, Calc. Var. Partial Differential Equations, 18 (2003), 117-143.  doi: 10.1007/s00526-002-0182-9.

[13]

P. Gerard, Description du défaut de compacité de l'injection de Sobolev, ESAIM Control Optim. Calc. Var., 3 (1998), 213-233.  doi: 10.1051/cocv:1998107.

[14]

S. Jaffard, Analysis of the lack of compactness in the critical Sobolev embeddings, J. Funct. Anal., 161 (1999), 384-396.  doi: 10.1006/jfan.1998.3364.

[15]

C. Mercuri, B. Sciunzi and M. Squassina, On Coron's problem for the $p$-Laplacian, J. Math. Anal. Appl., 421 (2015), 362-369.  doi: 10.1016/j.jmaa.2014.07.018.

[16]

C. Mercuri and M. Willem, A global compactness result for the $p$-Laplacian involving critical nonlinearities, Discrete Cont. Dyn. Syst., 28 (2010), 469-493.  doi: 10.3934/dcds.2010.28.469.

[17]

G. Palatucci and A. Pisante, Improved Sobolev embeddings, profile decomposition, and concentration compactness for fractional Sobolev spaces, Calc. Var. Partial Differential Equations, 50 (2014), 799-829.  doi: 10.1007/s00526-013-0656-y.

[18]

G. Palatucci and A. Pisante, A global compactness type result for Palais-Smale sequences in fractional Sobolev spaces, Nonlinear Anal., 117 (2015), 1-7.  doi: 10.1016/j.na.2014.12.027.

[19]

S. Secchi, N. Shioji and M. Squassina, Coron problem for fractional equations, Differential Integral Equations, 28 (2015), 103-118. 

[20]

W. Sickel, L. Skrzypczak and J. Vybiral, On the interplay of regularity and decay in case of radial functions Ⅰ. Inhomogeneous spaces Commun. Contemp. Math. 14 (2012), 1250005, 60 pp. doi: 10.1142/S0219199712500058.

[21]

M. Struwe, A global compactness result for elliptic boundary value problems involving limiting nonlinearities, Math. Z., 187 (1984), 511-517.  doi: 10.1007/BF01174186.

[22]

H. Triebel, Theory of Function Spaces. III, Monographs in Mathematics, 100. Birkhäuser Verlag, Basel, 2006.

[23]

H. Triebel, Theory of Function Spaces [Reprint of 1983 edition]. Modern Birkhäuser Classics. Birkhäuser/Springer Basel AG, Basel, 2010.

[24]

M. Willem, Minimax Theorems, Progress Nonlinear Differential Equations Appl. 24. Birkhäuser Boston, Inc., Boston, MA, 1996. doi: 10.1007/978-1-4612-4146-1.

[25]

S. Yan, A global compactness result for quasilinear elliptic equations with critical Sobolev exponents, Chinese Ann. Math. Ser. A, 16 (1995), 397-402. 

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