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October  2019, 39(10): 6149-6173. doi: 10.3934/dcds.2019268

Sign-changing bubble-tower solutions to fractional semilinear elliptic problems

Gabriele Cora 1, and  Alessandro Iacopetti 2,

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

Received  April 2019 Revised  April 2019 Published  July 2019

We study the asymptotic and qualitative properties of least energy radial sign-changing solutions to fractional semilinear elliptic problems of the form
$ \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} $
where
$ s \in (0,1) $
,
$ (-\Delta)^s $
 is the s-Laplacian,
$ B_R $
 is a ball of
$ \mathbb{R}^n $
,
$ 2^*_s : = \frac{2n}{n-2s} $
 is the critical Sobolev exponent and
$ \varepsilon>0 $
 is a small parameter. We prove that such solutions have the limit profile of a "tower of bubbles", as
$ \varepsilon \to 0^+ $
, i.e. the positive and negative parts concentrate at the same point with different concentration speeds. Moreover, we provide information about the nodal set of these solutions.
Keywords: Fractional semilinear elliptic equations, critical exponent, nodal regions, sign-changing radial solutions, asymptotic behavior.
Mathematics Subject Classification: Primary: 35J61, 35B40; Secondary: 35B44, 35B05.
Citation: Gabriele Cora, Alessandro Iacopetti. Sign-changing bubble-tower solutions to fractional semilinear elliptic problems. Discrete and Continuous Dynamical Systems, 2019, 39 (10) : 6149-6173. doi: 10.3934/dcds.2019268
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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