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Stability of normalized solitary waves for three coupled nonlinear Schrödinger equations
Stability of variational eigenvalues for the fractional $p-$Laplacian
1. | Aix-Marseille Université, CNRS, Centrale Marseille, I2M, UMR 7373, 39 Rue Frédéric Joliot Curie, 13453 Marseille, France, France |
2. | Dipartimento di Informatica, Università di Verona, Strada Le Grazie 15, 37134 Verona |
References:
[1] |
L. Ambrosio, G. De Philippis and L. Martinazzi, $\Gamma$-convergence of nonlocal perimeter functionals, Manuscripta Math., 134 (2011), 377-403.
doi: 10.1007/s00229-010-0399-4. |
[2] |
D. Applebaum, Lévy processes - from probability to finance and quantum groups, Notices Amer. Math. Soc., 51 (2004), 1336-1347. |
[3] |
J. Bourgain, H. Brezis and P. Mironescu, Another look at Sobolev spaces, in Optimal Control and Partial Differential Equations. A Volume in Honor of Professor Alain Bensoussan's 60th Birthday (eds. J. L. Menaldi, E. Rofman and A. Sulem), IOS Press, Amsterdam, 2001, 439-455. |
[4] |
J. Bourgain, H. Brezis and P. Mironescu, Limiting embedding theorems for $W^{s,p}$ when $s \to 1$ and applications, J. Anal. Math., 87 (2002), 77-101.
doi: 10.1007/BF02868470. |
[5] |
H. Brezis, Analyse Fonctionnelle: Théorie et Applications, Masson, Paris, 1983. |
[6] |
L. Brasco, E. Lindgren and E. Parini, The fractional Cheeger problem, Interfaces Free Bound., 16 (2014), 419-458.
doi: 10.4171/IFB/325. |
[7] |
L. Brasco and E. Parini, The second eigenvalue of the fractional $p$-Laplacian, to appear on Adv. Calc. Var. (2015)
doi: 10.1515/acv-2015-0007. |
[8] |
X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 23-53.
doi: 10.1016/j.anihpc.2013.02.001. |
[9] |
L. A. Caffarelli, Nonlocal equations, drifts and games, Nonlinear partial differential equations, Abel Symposia, 7 (2012), 37-52.
doi: 10.1007/978-3-642-25361-4_3. |
[10] |
T. Champion and L. De Pascale, Asymptotic behaviour of nonlinear eigenvalue problems involving $p$-Laplacian-type operators, Proc. Roy. Soc. Edinburgh Sect. A, 137 (2007), 1179-1195.
doi: 10.1017/S0308210506000667. |
[11] |
M. Cuesta, D. G. De Figueiredo and J.-P. Gossez, The beginning of the Fučik spectrum for the $p$-Laplacian, J. Differential Equations, 159 (1999), 212-238.
doi: 10.1006/jdeq.1999.3645. |
[12] |
G. Dal Maso, An Introduction to $\Gamma$-convergence, Progress in Nonlinear Differential Equations and their Applications, 8, Birkhäuser, Boston, 1993.
doi: 10.1007/978-1-4612-0327-8. |
[13] |
M. Degiovanni and M. Marzocchi, Limit of minimax values under $\Gamma$-convergence, Electron. J. Differential Equations, (2014), 19pp. |
[14] |
A. Di Castro, T. Kuusi and G. Palatucci, Local behavior of fractional $p$-minimizers, to appear on Ann. Inst. H. Poincaré Anal. Non Linéaire (2015)
doi: 10.1016/j.anihpc.2015.04.003. |
[15] |
B. Dyda, A fractional order Hardy inequality, Illinois J. Math., 48 (2004), 575-588. |
[16] |
L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1992. |
[17] |
E. R. Fadell and P. H. Rabinowitz, Bifurcation for odd potential operators and an alternative topological index, J. Funct. Anal., 26 (1977), 48-67.
doi: 10.1016/0022-1236(77)90015-5. |
[18] |
A. Fiscella, R. Servadei and E. Valdinoci, Density properties for fractional Sobolev spaces, Ann. Acad. Sci. Fenn. Math., 40 (2015), 235-253.
doi: 10.5186/aasfm.2015.4009. |
[19] |
I. Fonseca and G. Leoni, Modern Methods in the Calculus of Variations: $L^p$ Spaces, Springer Monographs in Mathematics. Springer, New York, 2007. |
[20] |
I. Fonseca and S. Müller, Quasi-convex integrands and lower semicontinuity in $L^1$, SIAM J. Math. Anal., 23 (1992), 1081-1098.
doi: 10.1137/0523060. |
[21] |
G. Franzina and G. Palatucci, Fractional $p$-eigenvalues, Riv. Mat. Univ. Parma, 5 (2014), 373-386. |
[22] |
E. Giusti, Direct Methods in the Calculus of Variations, World Scientific Publishing Co., Inc., River Edge, NJ, 2003.
doi: 10.1142/9789812795557. |
[23] |
A. Henrot, Extremum Problems for Eigenvalues of Elliptic Operators, Frontiers in Mathematics. Birkhäuser Verlag, Basel, 2006. |
[24] |
Y. X. Huang, On the eigenvalue of the $p$-Laplacian with varying $p$, Proc. Amer. Math. Soc., 125 (1997), 3347-3354.
doi: 10.1090/S0002-9939-97-03961-0. |
[25] |
A. Iannizzotto, S. Liu, K. Perera and M. Squassina, Existence results for fractional $p$-Laplacian problems via Morse theory, to appear on Adv. Calc. Var., (2015)
doi: 10.1515/acv-2014-0024. |
[26] |
A. Iannizzotto, S. Mosconi and M. Squassina, Global Hölder regularity for the fractional $p$-Laplacian, to appear on Rev. Mat. Iberoam.,, available at , ().
|
[27] |
A. Iannizzotto and M. Squassina, Weyl-type laws for fractional $p$-eigenvalue problems, Asymptot. Anal., 88 (2014), 233-245. |
[28] |
T. Kuusi, G. Mingione and Y. Sire, Nonlocal equations with measure data, Comm. Math. Phys., 337 (2015), 1317-1368.
doi: 10.1007/s00220-015-2356-2. |
[29] |
T. Kuusi, G. Mingione and Y. Sire, Nonlocal self-improving properties, Anal. PDE, 8 (2015), 57-114.
doi: 10.2140/apde.2015.8.57. |
[30] |
E. Lindgren and P. Lindqvist, Fractional eigenvalues, Calc. Var. Partial Differential Equations, 49 (2014), 795-826.
doi: 10.1007/s00526-013-0600-1. |
[31] |
P. Lindqvist, On non-linear Rayleigh quotients, Potential Anal., 2 (1993), 199-218.
doi: 10.1007/BF01048505. |
[32] |
S. Littig and F. Schuricht, Convergence of the eigenvalues of the $p$-Laplace operator as $p$ goes to 1, Calc. Var. Partial Differential Equations, 49 (2014), 707-727.
doi: 10.1007/s00526-013-0597-5. |
[33] |
M. Loss and C. Sloane, Hardy inequalities for fractional integrals on general domains, J. Funct. Anal., 259 (2010), 1369-1379.
doi: 10.1016/j.jfa.2010.05.001. |
[34] |
V. Maz'ya and T. Shaposhnikova, On the Bourgain, Brezis, and Mironescu theorem concerning limiting embeddings of fractional Sobolev spaces, J. Funct. Anal., 195 (2002), 230-238.
doi: 10.1006/jfan.2002.3955. |
[35] |
E. Parini, Continuity of the variational eigenvalues of the $p$-Laplacian with respect to $p$, Bull. Aust. Math. Soc., 83 (2011), 376-381.
doi: 10.1017/S000497271100205X. |
[36] |
A. Ponce, A new approach to Sobolev spaces and connections to $\Gamma$-convergence, Calc. Var. Partial Differential Equations, 19 (2004), 229-255.
doi: 10.1007/s00526-003-0195-z. |
[37] |
Y. Sire and E. Valdinoci, Rigidity results for some boundary quasilinear phase transitions, Commun. Partial Differ. Equations, 34 (2009), 765-784.
doi: 10.1080/03605300902892402. |
[38] |
M. Struwe, Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, forth edition, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, 34. Springer-Verlag, Berlin, 2008. |
show all references
References:
[1] |
L. Ambrosio, G. De Philippis and L. Martinazzi, $\Gamma$-convergence of nonlocal perimeter functionals, Manuscripta Math., 134 (2011), 377-403.
doi: 10.1007/s00229-010-0399-4. |
[2] |
D. Applebaum, Lévy processes - from probability to finance and quantum groups, Notices Amer. Math. Soc., 51 (2004), 1336-1347. |
[3] |
J. Bourgain, H. Brezis and P. Mironescu, Another look at Sobolev spaces, in Optimal Control and Partial Differential Equations. A Volume in Honor of Professor Alain Bensoussan's 60th Birthday (eds. J. L. Menaldi, E. Rofman and A. Sulem), IOS Press, Amsterdam, 2001, 439-455. |
[4] |
J. Bourgain, H. Brezis and P. Mironescu, Limiting embedding theorems for $W^{s,p}$ when $s \to 1$ and applications, J. Anal. Math., 87 (2002), 77-101.
doi: 10.1007/BF02868470. |
[5] |
H. Brezis, Analyse Fonctionnelle: Théorie et Applications, Masson, Paris, 1983. |
[6] |
L. Brasco, E. Lindgren and E. Parini, The fractional Cheeger problem, Interfaces Free Bound., 16 (2014), 419-458.
doi: 10.4171/IFB/325. |
[7] |
L. Brasco and E. Parini, The second eigenvalue of the fractional $p$-Laplacian, to appear on Adv. Calc. Var. (2015)
doi: 10.1515/acv-2015-0007. |
[8] |
X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 23-53.
doi: 10.1016/j.anihpc.2013.02.001. |
[9] |
L. A. Caffarelli, Nonlocal equations, drifts and games, Nonlinear partial differential equations, Abel Symposia, 7 (2012), 37-52.
doi: 10.1007/978-3-642-25361-4_3. |
[10] |
T. Champion and L. De Pascale, Asymptotic behaviour of nonlinear eigenvalue problems involving $p$-Laplacian-type operators, Proc. Roy. Soc. Edinburgh Sect. A, 137 (2007), 1179-1195.
doi: 10.1017/S0308210506000667. |
[11] |
M. Cuesta, D. G. De Figueiredo and J.-P. Gossez, The beginning of the Fučik spectrum for the $p$-Laplacian, J. Differential Equations, 159 (1999), 212-238.
doi: 10.1006/jdeq.1999.3645. |
[12] |
G. Dal Maso, An Introduction to $\Gamma$-convergence, Progress in Nonlinear Differential Equations and their Applications, 8, Birkhäuser, Boston, 1993.
doi: 10.1007/978-1-4612-0327-8. |
[13] |
M. Degiovanni and M. Marzocchi, Limit of minimax values under $\Gamma$-convergence, Electron. J. Differential Equations, (2014), 19pp. |
[14] |
A. Di Castro, T. Kuusi and G. Palatucci, Local behavior of fractional $p$-minimizers, to appear on Ann. Inst. H. Poincaré Anal. Non Linéaire (2015)
doi: 10.1016/j.anihpc.2015.04.003. |
[15] |
B. Dyda, A fractional order Hardy inequality, Illinois J. Math., 48 (2004), 575-588. |
[16] |
L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1992. |
[17] |
E. R. Fadell and P. H. Rabinowitz, Bifurcation for odd potential operators and an alternative topological index, J. Funct. Anal., 26 (1977), 48-67.
doi: 10.1016/0022-1236(77)90015-5. |
[18] |
A. Fiscella, R. Servadei and E. Valdinoci, Density properties for fractional Sobolev spaces, Ann. Acad. Sci. Fenn. Math., 40 (2015), 235-253.
doi: 10.5186/aasfm.2015.4009. |
[19] |
I. Fonseca and G. Leoni, Modern Methods in the Calculus of Variations: $L^p$ Spaces, Springer Monographs in Mathematics. Springer, New York, 2007. |
[20] |
I. Fonseca and S. Müller, Quasi-convex integrands and lower semicontinuity in $L^1$, SIAM J. Math. Anal., 23 (1992), 1081-1098.
doi: 10.1137/0523060. |
[21] |
G. Franzina and G. Palatucci, Fractional $p$-eigenvalues, Riv. Mat. Univ. Parma, 5 (2014), 373-386. |
[22] |
E. Giusti, Direct Methods in the Calculus of Variations, World Scientific Publishing Co., Inc., River Edge, NJ, 2003.
doi: 10.1142/9789812795557. |
[23] |
A. Henrot, Extremum Problems for Eigenvalues of Elliptic Operators, Frontiers in Mathematics. Birkhäuser Verlag, Basel, 2006. |
[24] |
Y. X. Huang, On the eigenvalue of the $p$-Laplacian with varying $p$, Proc. Amer. Math. Soc., 125 (1997), 3347-3354.
doi: 10.1090/S0002-9939-97-03961-0. |
[25] |
A. Iannizzotto, S. Liu, K. Perera and M. Squassina, Existence results for fractional $p$-Laplacian problems via Morse theory, to appear on Adv. Calc. Var., (2015)
doi: 10.1515/acv-2014-0024. |
[26] |
A. Iannizzotto, S. Mosconi and M. Squassina, Global Hölder regularity for the fractional $p$-Laplacian, to appear on Rev. Mat. Iberoam.,, available at , ().
|
[27] |
A. Iannizzotto and M. Squassina, Weyl-type laws for fractional $p$-eigenvalue problems, Asymptot. Anal., 88 (2014), 233-245. |
[28] |
T. Kuusi, G. Mingione and Y. Sire, Nonlocal equations with measure data, Comm. Math. Phys., 337 (2015), 1317-1368.
doi: 10.1007/s00220-015-2356-2. |
[29] |
T. Kuusi, G. Mingione and Y. Sire, Nonlocal self-improving properties, Anal. PDE, 8 (2015), 57-114.
doi: 10.2140/apde.2015.8.57. |
[30] |
E. Lindgren and P. Lindqvist, Fractional eigenvalues, Calc. Var. Partial Differential Equations, 49 (2014), 795-826.
doi: 10.1007/s00526-013-0600-1. |
[31] |
P. Lindqvist, On non-linear Rayleigh quotients, Potential Anal., 2 (1993), 199-218.
doi: 10.1007/BF01048505. |
[32] |
S. Littig and F. Schuricht, Convergence of the eigenvalues of the $p$-Laplace operator as $p$ goes to 1, Calc. Var. Partial Differential Equations, 49 (2014), 707-727.
doi: 10.1007/s00526-013-0597-5. |
[33] |
M. Loss and C. Sloane, Hardy inequalities for fractional integrals on general domains, J. Funct. Anal., 259 (2010), 1369-1379.
doi: 10.1016/j.jfa.2010.05.001. |
[34] |
V. Maz'ya and T. Shaposhnikova, On the Bourgain, Brezis, and Mironescu theorem concerning limiting embeddings of fractional Sobolev spaces, J. Funct. Anal., 195 (2002), 230-238.
doi: 10.1006/jfan.2002.3955. |
[35] |
E. Parini, Continuity of the variational eigenvalues of the $p$-Laplacian with respect to $p$, Bull. Aust. Math. Soc., 83 (2011), 376-381.
doi: 10.1017/S000497271100205X. |
[36] |
A. Ponce, A new approach to Sobolev spaces and connections to $\Gamma$-convergence, Calc. Var. Partial Differential Equations, 19 (2004), 229-255.
doi: 10.1007/s00526-003-0195-z. |
[37] |
Y. Sire and E. Valdinoci, Rigidity results for some boundary quasilinear phase transitions, Commun. Partial Differ. Equations, 34 (2009), 765-784.
doi: 10.1080/03605300902892402. |
[38] |
M. Struwe, Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, forth edition, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, 34. Springer-Verlag, Berlin, 2008. |
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