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

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

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.

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. BrascoE. 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 CastroT. 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 NezzaG. 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. GazzolaH. 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. MercuriB. 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. SecchiN. 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. BrascoE. 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 CastroT. 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 NezzaG. 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. GazzolaH. 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. MercuriB. 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. SecchiN. 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. 

[1]

Pavel Jirásek. On Compactness Conditions for the $p$-Laplacian. Communications on Pure and Applied Analysis, 2016, 15 (3) : 715-726. doi: 10.3934/cpaa.2016.15.715

[2]

Carlo Mercuri, Michel Willem. A global compactness result for the p-Laplacian involving critical nonlinearities. Discrete and Continuous Dynamical Systems, 2010, 28 (2) : 469-493. doi: 10.3934/dcds.2010.28.469

[3]

Lorenzo Brasco, Enea Parini, Marco Squassina. Stability of variational eigenvalues for the fractional $p-$Laplacian. Discrete and Continuous Dynamical Systems, 2016, 36 (4) : 1813-1845. doi: 10.3934/dcds.2016.36.1813

[4]

Hugo Beirão da Veiga, Francesca Crispo. On the global regularity for nonlinear systems of the $p$-Laplacian type. Discrete and Continuous Dynamical Systems - S, 2013, 6 (5) : 1173-1191. doi: 10.3934/dcdss.2013.6.1173

[5]

Yixuan Wu, Yanzhi Zhang. Highly accurate operator factorization methods for the integral fractional Laplacian and its generalization. Discrete and Continuous Dynamical Systems - S, 2022, 15 (4) : 851-876. doi: 10.3934/dcdss.2022016

[6]

Patrizia Pucci, Mingqi Xiang, Binlin Zhang. A diffusion problem of Kirchhoff type involving the nonlocal fractional p-Laplacian. Discrete and Continuous Dynamical Systems, 2017, 37 (7) : 4035-4051. doi: 10.3934/dcds.2017171

[7]

Lingyu Jin, Yan Li. A Hopf's lemma and the boundary regularity for the fractional p-Laplacian. Discrete and Continuous Dynamical Systems, 2019, 39 (3) : 1477-1495. doi: 10.3934/dcds.2019063

[8]

CÉSAR E. TORRES LEDESMA. Existence and symmetry result for fractional p-Laplacian in $\mathbb{R}^{n}$. Communications on Pure and Applied Analysis, 2017, 16 (1) : 99-114. doi: 10.3934/cpaa.2017004

[9]

Leyun Wu, Pengcheng Niu. Symmetry and nonexistence of positive solutions to fractional p-Laplacian equations. Discrete and Continuous Dynamical Systems, 2019, 39 (3) : 1573-1583. doi: 10.3934/dcds.2019069

[10]

Jacson Simsen, José Valero. Global attractors for $p$-Laplacian differential inclusions in unbounded domains. Discrete and Continuous Dynamical Systems - B, 2016, 21 (9) : 3239-3267. doi: 10.3934/dcdsb.2016096

[11]

Tomás Caraballo, Marta Herrera-Cobos, Pedro Marín-Rubio. Global attractor for a nonlocal p-Laplacian equation without uniqueness of solution. Discrete and Continuous Dynamical Systems - B, 2017, 22 (5) : 1801-1816. doi: 10.3934/dcdsb.2017107

[12]

Mohammad A. Rammaha, Daniel Toundykov, Zahava Wilstein. Global existence and decay of energy for a nonlinear wave equation with $p$-Laplacian damping. Discrete and Continuous Dynamical Systems, 2012, 32 (12) : 4361-4390. doi: 10.3934/dcds.2012.32.4361

[13]

Guowei Dai, Ruyun Ma. Unilateral global bifurcation for $p$-Laplacian with non-$p-$1-linearization nonlinearity. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 99-116. doi: 10.3934/dcds.2015.35.99

[14]

Hantaek Bae. On the local and global existence of the Hall equations with fractional Laplacian and related equations. Networks and Heterogeneous Media, 2022  doi: 10.3934/nhm.2022021

[15]

Yong Zhou, V. Vijayakumar, R. Murugesu. Controllability for fractional evolution inclusions without compactness. Evolution Equations and Control Theory, 2015, 4 (4) : 507-524. doi: 10.3934/eect.2015.4.507

[16]

Phuong Le. Symmetry of singular solutions for a weighted Choquard equation involving the fractional $ p $-Laplacian. Communications on Pure and Applied Analysis, 2020, 19 (1) : 527-539. doi: 10.3934/cpaa.2020026

[17]

Vincenzo Ambrosio, Teresa Isernia. Multiplicity and concentration results for some nonlinear Schrödinger equations with the fractional p-Laplacian. Discrete and Continuous Dynamical Systems, 2018, 38 (11) : 5835-5881. doi: 10.3934/dcds.2018254

[18]

Zaizheng Li, Qidi Zhang. Sub-solutions and a point-wise Hopf's lemma for fractional $ p $-Laplacian. Communications on Pure and Applied Analysis, 2021, 20 (2) : 835-865. doi: 10.3934/cpaa.2020293

[19]

Elhoussine Azroul, Abdelmoujib Benkirane, and Mohammed Shimi. On a nonlocal problem involving the fractional $ p(x,.) $-Laplacian satisfying Cerami condition. Discrete and Continuous Dynamical Systems - S, 2021, 14 (10) : 3479-3495. doi: 10.3934/dcdss.2020425

[20]

Lihong Zhang, Wenwen Hou, Bashir Ahmad, Guotao Wang. Radial symmetry for logarithmic Choquard equation involving a generalized tempered fractional $ p $-Laplacian. Discrete and Continuous Dynamical Systems - S, 2021, 14 (10) : 3851-3863. doi: 10.3934/dcdss.2020445

2020 Impact Factor: 2.425

Metrics

  • PDF downloads (197)
  • HTML views (211)
  • Cited by (6)

Other articles
by authors

[Back to Top]