American Institute of Mathematical Sciences

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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 & 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.  Google Scholar

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[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.  Google Scholar

[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.  Google Scholar

[22]

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

[23]

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

[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.  Google Scholar

[25]

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

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.  Google Scholar

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[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.  Google Scholar

[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.  Google Scholar

[22]

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

[23]

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

[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.  Google Scholar

[25]

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

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