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On a new fractional Sobolev space and applications to nonlocal variational problems with variable exponent
Global compactness results for nonlocal problems
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 |
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.
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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