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Accelerating dynamical peakons and their behaviour
Sign-changing bubble-tower solutions to fractional semilinear elliptic problems
1. | Dipartimento di Matematica "G. Peano", Università di Torino, via Carlo Alberto 10 – 10123 Torino, Italy |
2. | Dipartimento di Matematica, Università di Roma "La Sapienza", P.le Aldo Moro 5 – 00185 Roma, Italy |
$ \begin{cases} (-\Delta)^s u = |u|^{2^*_s-2-\varepsilon}u &\text{in } B_R, \\ u = 0 &\text{in } \mathbb{R}^n \setminus B_R, \end{cases} $ |
$ s \in (0,1) $ |
$ (-\Delta)^s $ |
$ B_R $ |
$ \mathbb{R}^n $ |
$ 2^*_s : = \frac{2n}{n-2s} $ |
$ \varepsilon>0 $ |
$ \varepsilon \to 0^+ $ |
References:
[1] |
F. V. Atkinson, H. Brezis and L. A. Peletier,
Solutions d'equations elliptiques avec exposant de Sobolev critique qui changent de signe, C. R. Acad. Sci. Paris Sér. I Math., 306 (1988), 711-714.
|
[2] |
M. Ben Ayed, K. El Mehdi and F. Pacella,
Blow-up and symmetry of sign-changing solutions to some critical elliptic equations, J. Differential Equations, 230 (2006), 771-795.
doi: 10.1016/j.jde.2006.05.008. |
[3] |
M. Ben Ayed, K. El Mehdi and F. Pacella,
Classification of low energy sign-changing solutions of an almost critical problem, J. Funct. Anal., 250 (2007), 347-373.
doi: 10.1016/j.jfa.2007.05.024. |
[4] |
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. |
[5] |
W. Chen, C. Li and Y. Li,
A direct method of moving planes for the fractional Laplacian, Adv. Math., 308 (2017), 404-437.
doi: 10.1016/j.aim.2016.11.038. |
[6] |
Y. Cho and T. Ozawa,
Sobolev inequalities with symmetry, Commun. Contemp. Math., 11 (2009), 355-365.
doi: 10.1142/S0219199709003399. |
[7] |
W. Choi, S. Kim and K.-A. Lee,
Asymptotic behavior for solutions for nonlinear elliptic problems with the fractional Laplacian, J. Funct. Anal., 266 (2014), 6531-6598.
doi: 10.1016/j.jfa.2014.02.029. |
[8] |
G. Cora and A. Iacopetti,
On the structure of the nodal set and asymptotics of least energy sign-changing radial solutions of the fractional Brezis-Nirenberg problem, Nonlinear Anal., 176 (2018), 226-271.
doi: 10.1016/j.na.2018.07.001. |
[9] |
A. Cotsiolis and N. Tavoularis,
Best constants for Sobolev inequalities for higher order fractional derivatives, J. Math. Anal. Appl., 295 (2004), 225-236.
doi: 10.1016/j.jmaa.2004.03.034. |
[10] |
M. del Pino, J. Dolbeault and M. Musso,
Bubble-tower radial solutions in the slightly supercritical Brezis-Nirenberg problem, J. Differential Equations, 193 (2003), 280-306.
doi: 10.1016/S0022-0396(03)00151-7. |
[11] |
E. Di Nezza, G. Palatucci and E. Valdinoci,
Hitchhicker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.
doi: 10.1016/j.bulsci.2011.12.004. |
[12] |
M. M. Fall and E. Valdinoci,
Uniqueness and Nondegeneracy of Positive Solutions of $(-\Delta)^s u+u = u^p$ in $\mathbb{R}^N$ when s is Close to 1, Comm. Math. Phys., 329 (2014), 383-404.
doi: 10.1007/s00220-014-1919-y. |
[13] |
M. M. Fall and V. Felli,
Unique continuation property and local asymptotics of solutions to fractional elliptic equations, Comm. Partial Differential Equations, 39 (2014), 354-397.
doi: 10.1080/03605302.2013.825918. |
[14] |
A. Farina,
On the classification of solutions of the Lane-Emden equation on unbounded domains of $\mathbb{R}^n$, J. Math. Pures Appl., 87 (2007), 537-561.
doi: 10.1016/j.matpur.2007.03.001. |
[15] |
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. |
[16] |
R. L. Frank and E. Lenzmann,
Uniqueness of non-linear ground states for fractional Laplacians in $\mathbb{R}$, Acta Math., 210 (2013), 261-318.
doi: 10.1007/s11511-013-0095-9. |
[17] |
R. L. Frank, E. Lenzmann and L. Silvestre,
Uniqueness of radial solutions for the fractional Laplacian, Comm. Pure Appl. Math., 69 (2016), 1671-1726.
doi: 10.1002/cpa.21591. |
[18] |
Z.-C. Han,
Asymptotic approach to singular solutions for nonlinear elliptic equations involving critical Sobolev exponent, Ann. Inst. H. Poincaré Anal. Non Linéaire, 8 (1991), 159-174.
doi: 10.1016/S0294-1449(16)30270-0. |
[19] |
A. Iacopetti and F. Pacella,
A nonexistence result for sign-changing solutions of the Brezis-Nirenberg problem in low dimensions, J. Differential Equations, 258 (2015), 4180-4208.
doi: 10.1016/j.jde.2015.01.030. |
[20] |
A. Iacopetti and G. Vaira, Sign-changing tower of bubbles for the Brezis-Nirenberg problem, Commun. Contemp. Math., 18 (2016), 1550036, 53pp.
doi: 10.1142/S0219199715500364. |
[21] |
A. Iacopetti and G. Vaira,
Sign-changing blowing-up solutions for the Brezis–Nirenberg problem in dimensions four and five, Ann. Sc. Norm. Super. Pisa Cl. Sci., 18 (2018), 1-38.
|
[22] |
A. Iannizzotto, S. Mosconi and M. Squassina,
$H^s$ versus $C^0$-weighted minimizers, Nonlinear Differential Equations Appl., 22 (2015), 477-497.
doi: 10.1007/s00030-014-0292-z. |
[23] |
A. Pistoia and T. Weth,
Sign-changing bubble tower solutions in a slightly subcritical semilinear Dirichlet problem, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 325-340.
doi: 10.1016/j.anihpc.2006.03.002. |
[24] |
O. Rey,
Proof of two conjectures of H. Brezis and L. A. Peletier, Manuscripta Math., 65 (1989), 19-37.
doi: 10.1007/BF01168364. |
[25] |
X. Ros-Oton and J. Serra,
The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary, J. Math. Pures Appl., 101 (2014), 275-302.
doi: 10.1016/j.matpur.2013.06.003. |
[26] |
X. Ros-Oton and J. Serra,
The Pohozaev identity for the fractional Laplacian, Arch. Ration. Mech. Anal., 213 (2014), 587-628.
doi: 10.1007/s00205-014-0740-2. |
[27] |
R. Servadei,
The Yamabe equation in a non-local setting, Adv. Nonlinear Anal., 2 (2013), 235-270.
doi: 10.1515/anona-2013-0008. |
[28] |
R. Servadei and E. Valdinoci,
The Brezis-Nirenberg result for the fractional Laplacian, Trans. Amer. Math. Soc., 367 (2015), 67-102.
doi: 10.1090/S0002-9947-2014-05884-4. |
[29] |
K. Teng, K. Wang and R. Wang,
A sign-changing solution for nonlinear problems involving the fractional Laplacian, Electron. J. Differential Equations, 109 (2015), 1-12.
|
show all references
References:
[1] |
F. V. Atkinson, H. Brezis and L. A. Peletier,
Solutions d'equations elliptiques avec exposant de Sobolev critique qui changent de signe, C. R. Acad. Sci. Paris Sér. I Math., 306 (1988), 711-714.
|
[2] |
M. Ben Ayed, K. El Mehdi and F. Pacella,
Blow-up and symmetry of sign-changing solutions to some critical elliptic equations, J. Differential Equations, 230 (2006), 771-795.
doi: 10.1016/j.jde.2006.05.008. |
[3] |
M. Ben Ayed, K. El Mehdi and F. Pacella,
Classification of low energy sign-changing solutions of an almost critical problem, J. Funct. Anal., 250 (2007), 347-373.
doi: 10.1016/j.jfa.2007.05.024. |
[4] |
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. |
[5] |
W. Chen, C. Li and Y. Li,
A direct method of moving planes for the fractional Laplacian, Adv. Math., 308 (2017), 404-437.
doi: 10.1016/j.aim.2016.11.038. |
[6] |
Y. Cho and T. Ozawa,
Sobolev inequalities with symmetry, Commun. Contemp. Math., 11 (2009), 355-365.
doi: 10.1142/S0219199709003399. |
[7] |
W. Choi, S. Kim and K.-A. Lee,
Asymptotic behavior for solutions for nonlinear elliptic problems with the fractional Laplacian, J. Funct. Anal., 266 (2014), 6531-6598.
doi: 10.1016/j.jfa.2014.02.029. |
[8] |
G. Cora and A. Iacopetti,
On the structure of the nodal set and asymptotics of least energy sign-changing radial solutions of the fractional Brezis-Nirenberg problem, Nonlinear Anal., 176 (2018), 226-271.
doi: 10.1016/j.na.2018.07.001. |
[9] |
A. Cotsiolis and N. Tavoularis,
Best constants for Sobolev inequalities for higher order fractional derivatives, J. Math. Anal. Appl., 295 (2004), 225-236.
doi: 10.1016/j.jmaa.2004.03.034. |
[10] |
M. del Pino, J. Dolbeault and M. Musso,
Bubble-tower radial solutions in the slightly supercritical Brezis-Nirenberg problem, J. Differential Equations, 193 (2003), 280-306.
doi: 10.1016/S0022-0396(03)00151-7. |
[11] |
E. Di Nezza, G. Palatucci and E. Valdinoci,
Hitchhicker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.
doi: 10.1016/j.bulsci.2011.12.004. |
[12] |
M. M. Fall and E. Valdinoci,
Uniqueness and Nondegeneracy of Positive Solutions of $(-\Delta)^s u+u = u^p$ in $\mathbb{R}^N$ when s is Close to 1, Comm. Math. Phys., 329 (2014), 383-404.
doi: 10.1007/s00220-014-1919-y. |
[13] |
M. M. Fall and V. Felli,
Unique continuation property and local asymptotics of solutions to fractional elliptic equations, Comm. Partial Differential Equations, 39 (2014), 354-397.
doi: 10.1080/03605302.2013.825918. |
[14] |
A. Farina,
On the classification of solutions of the Lane-Emden equation on unbounded domains of $\mathbb{R}^n$, J. Math. Pures Appl., 87 (2007), 537-561.
doi: 10.1016/j.matpur.2007.03.001. |
[15] |
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. |
[16] |
R. L. Frank and E. Lenzmann,
Uniqueness of non-linear ground states for fractional Laplacians in $\mathbb{R}$, Acta Math., 210 (2013), 261-318.
doi: 10.1007/s11511-013-0095-9. |
[17] |
R. L. Frank, E. Lenzmann and L. Silvestre,
Uniqueness of radial solutions for the fractional Laplacian, Comm. Pure Appl. Math., 69 (2016), 1671-1726.
doi: 10.1002/cpa.21591. |
[18] |
Z.-C. Han,
Asymptotic approach to singular solutions for nonlinear elliptic equations involving critical Sobolev exponent, Ann. Inst. H. Poincaré Anal. Non Linéaire, 8 (1991), 159-174.
doi: 10.1016/S0294-1449(16)30270-0. |
[19] |
A. Iacopetti and F. Pacella,
A nonexistence result for sign-changing solutions of the Brezis-Nirenberg problem in low dimensions, J. Differential Equations, 258 (2015), 4180-4208.
doi: 10.1016/j.jde.2015.01.030. |
[20] |
A. Iacopetti and G. Vaira, Sign-changing tower of bubbles for the Brezis-Nirenberg problem, Commun. Contemp. Math., 18 (2016), 1550036, 53pp.
doi: 10.1142/S0219199715500364. |
[21] |
A. Iacopetti and G. Vaira,
Sign-changing blowing-up solutions for the Brezis–Nirenberg problem in dimensions four and five, Ann. Sc. Norm. Super. Pisa Cl. Sci., 18 (2018), 1-38.
|
[22] |
A. Iannizzotto, S. Mosconi and M. Squassina,
$H^s$ versus $C^0$-weighted minimizers, Nonlinear Differential Equations Appl., 22 (2015), 477-497.
doi: 10.1007/s00030-014-0292-z. |
[23] |
A. Pistoia and T. Weth,
Sign-changing bubble tower solutions in a slightly subcritical semilinear Dirichlet problem, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 325-340.
doi: 10.1016/j.anihpc.2006.03.002. |
[24] |
O. Rey,
Proof of two conjectures of H. Brezis and L. A. Peletier, Manuscripta Math., 65 (1989), 19-37.
doi: 10.1007/BF01168364. |
[25] |
X. Ros-Oton and J. Serra,
The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary, J. Math. Pures Appl., 101 (2014), 275-302.
doi: 10.1016/j.matpur.2013.06.003. |
[26] |
X. Ros-Oton and J. Serra,
The Pohozaev identity for the fractional Laplacian, Arch. Ration. Mech. Anal., 213 (2014), 587-628.
doi: 10.1007/s00205-014-0740-2. |
[27] |
R. Servadei,
The Yamabe equation in a non-local setting, Adv. Nonlinear Anal., 2 (2013), 235-270.
doi: 10.1515/anona-2013-0008. |
[28] |
R. Servadei and E. Valdinoci,
The Brezis-Nirenberg result for the fractional Laplacian, Trans. Amer. Math. Soc., 367 (2015), 67-102.
doi: 10.1090/S0002-9947-2014-05884-4. |
[29] |
K. Teng, K. Wang and R. Wang,
A sign-changing solution for nonlinear problems involving the fractional Laplacian, Electron. J. Differential Equations, 109 (2015), 1-12.
|
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