In this paper, we studied the fractional $ p $-Laplace problem with a general potential and a mass-critical nonlinearity. By using constrained variational methods, we established the existence and nonexistence of minimizers for this problem. Moreover, we investigated the blow-up behaviour of non-negative minimizers to the above equation. Finally, under a polynomial-type potential, we obtained the optimal rate of blow-up through some subtle energy estimates.
Citation: Xinyue Zhang, Haibo Chen, Jie Yang. Blow up behavior of minimizers for a fractional p-Laplace problem with external potentials and mass critical nonlinearity[J]. AIMS Mathematics, 2025, 10(2): 3597-3622. doi: 10.3934/math.2025166
In this paper, we studied the fractional $ p $-Laplace problem with a general potential and a mass-critical nonlinearity. By using constrained variational methods, we established the existence and nonexistence of minimizers for this problem. Moreover, we investigated the blow-up behaviour of non-negative minimizers to the above equation. Finally, under a polynomial-type potential, we obtained the optimal rate of blow-up through some subtle energy estimates.
[1] |
V. Ambrosio, On the Pohozaev identity for the fractional $p$-Laplacian operator in $\mathbb{R}^{N}$, B. Lond. Math. Soc., 56 (2024), 1999–2013. http://doi.org/10.1112/blms.13039 doi: 10.1112/blms.13039
![]() |
[2] |
V. Ambrosio, T. Isernia, Multiplicity and concentration results for some nonlinear Schrödinger equations with the fractional p-laplacian, DCDS, 38 (2018), 5835–5881. https://doi.org/10.3934/dcds.2018254 doi: 10.3934/dcds.2018254
![]() |
[3] |
V. Ambrosio, T. Isernia, V. D. Radulescu, Concentration of positive solutions for a class of fractional $p$-Kirchhoff type equations, P. Roy. Soc. Edinb. A, 151 (2021), 601–651. https://doi.org/10.1017/prm.2020.32 doi: 10.1017/prm.2020.32
![]() |
[4] |
L. Caffarelli, L. Silvestre, An extension problem related to the fractional Laplacian, Commun. Part. Diff. Eq., 32 (2007), 1245–1260. https://doi.org/10.1080/03605300600987306 doi: 10.1080/03605300600987306
![]() |
[5] |
A. Di Castro, T. Kuusi, G. Palatucci, Local behavior of fractional p-minimizers, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 1279–1299. https://doi.org/10.1016/j.anihpc.2015.04.003 doi: 10.1016/j.anihpc.2015.04.003
![]() |
[6] |
L. M. Del Pezzo, A. Quaas, Spectrum of the fractional p-Laplacian in $ \mathbb{R}^{N}$ and decay estimate for positive solutions of a Schrödinger equation, Nonlinear Anal., 193 (2020), 111479. https://doi.org/10.1016/j.na.2019.03.002 doi: 10.1016/j.na.2019.03.002
![]() |
[7] |
E. Di Nezza, G. Palatucci, E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, B. Sci. Math., 136 (2012), 521–573. https://doi.org/10.1016/j.bulsci.2011.12.004 doi: 10.1016/j.bulsci.2011.12.004
![]() |
[8] |
M. Du, L. X. Tian, J. Wang, F. B. Zhang, Existence of normalized solutions for nonlinear fractional Schrödinger equations with trapping potentials, P. Roy. Soc. Edinb. A, 149 (2019), 617–653. https://doi.org/10.1017/prm.2018.41 doi: 10.1017/prm.2018.41
![]() |
[9] |
P. Felmer, A. Quaas, J. Tan, Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian, P. Roy. Soc. Edinb. A, 142 (2012), 1237–1262. https://doi.org/10.1017/S0308210511000746 doi: 10.1017/S0308210511000746
![]() |
[10] |
E. P. Gross, Hydrodynamics of a superfluid condensate, J. Math. Phys., 4 (1963), 195–207. https://doi.org/10.1063/1.1703944 doi: 10.1063/1.1703944
![]() |
[11] |
Y. Guo, R. Seiringer, On the mass concentration for Bose–Einstein condensates with attractive interactions, Lett. Math. Phys., 104 (2014), 141–156. https://doi.org/10.1007/s11005-013-0667-9 doi: 10.1007/s11005-013-0667-9
![]() |
[12] |
A. Iannizzotto, S. J. N. Mosconi, M. Squassina, Global Hölder regularity for the fractional $p$-Laplacian, Rev. Mat. Iberoam., 32 (2016), 1353–1392. https://doi.org/10.4171/RMI/921 doi: 10.4171/RMI/921
![]() |
[13] |
S. Liu, H. Chen, Fractional Kirchhoff-type equation with singular potential and critical exponent, J. Math. Phys., 62 (2021), 111505. https://doi.org/10.1063/5.0061144 doi: 10.1063/5.0061144
![]() |
[14] |
Z. Liu, M. Squassina, J. Zhang, Ground states for fractional Kirchhoff equations with critical nonlinearity in low dimension, Nonlinear Differ. Equ. Appl., 24 (2017), 50. https://doi.org/10.1007/s00030-017-0473-7 doi: 10.1007/s00030-017-0473-7
![]() |
[15] |
L. Liu, K. Teng, J. Yang, H. Chen, Concentration behaviour of normalized ground states of the mass critical fractional Schrodinger equations with ring-shaped potentials, P. Roy. Soc. Edinb. A., 153 (2023), 1993–2024. https://doi.org/10.1017/prm.2022.81 doi: 10.1017/prm.2022.81
![]() |
[16] |
L. Liu, K. Teng, J. Yang, H. Chen, Properties of minimizers for the fractional Kirchhoff energy functional, J. Math. Phys., 64, (2023), 081504. https://doi.org/10.1063/5.0157267 doi: 10.1063/5.0157267
![]() |
[17] |
L. Liu, K. Teng, J. Yang, H. Chen, Minimizers of fractional NLS energy functionals in $ \mathbb{R}^{2}$, Comp. Appl. Math., 43 (2024), 23. https://doi.org/10.1007/s40314-023-02531-3 doi: 10.1007/s40314-023-02531-3
![]() |
[18] |
Q. Lou, Y. Qin, F. Liu, The existence of constrained minimizers related to fractional p-Laplacian equations, Topol. Method. Nonl. An., 58 (2021), 657–676. https://doi.org/10.12775/TMNA.2020.079 doi: 10.12775/TMNA.2020.079
![]() |
[19] |
M. Marin, On the minimum principle for dipolar materials with stretch, Nonlinear Anal.-Real, 10 (2009), 1572–1578. https://doi.org/10.1016/j.nonrwa.2008.02.001 doi: 10.1016/j.nonrwa.2008.02.001
![]() |
[20] | M. Marin, S. Gabriel, Weak solutions in Elasticity of dipolar bodies with stretch, Carpathian J. Math., 29 (2013), 33–40. |
[21] |
H. Nguyen, M. Squassina, Fractional Caffarelli-Kohn-Nirenberg inequalities, J. Funct. Anal., 274 (2018), 2661–2672. https://doi.org/10.1016/j.jfa.2017.07.007 doi: 10.1016/j.jfa.2017.07.007
![]() |
[22] |
T. V. Phan, Blow-up profile of Bose-Einstein condensate with singular potentials, J. Math. Phys., 58 (2017), 072301. https://doi.org/10.1063/1.4995393 doi: 10.1063/1.4995393
![]() |
[23] | L. P. Pitaevskii, Vortex lines in an imperfect Bose gas, Sov. Phys. JETP, 13 (1961), 451–454. |
[24] |
S. Secchi, Ground state solutions for nonlinear fractional Schrödinger equations in $ \mathbb{R}^{N}$, J. Math. Phys., 54 (2013), 031501. https://doi.org/10.1063/1.4793990 doi: 10.1063/1.4793990
![]() |
[25] |
Y. Su, Z. Liu, Semiclassical states to nonlinear Choquard equation with critical growth, Isr. J. Math., 255 (2023), 729–762. https://doi.org/10.1007/s11856-023-2485-9 doi: 10.1007/s11856-023-2485-9
![]() |
[26] |
Y. Su, Z. Liu, Semi-classical states for the Choquard equations with doubly critical exponents: Existence, multiplicity and concentration, Asymptotic Anal., 132 (2023), 451–493. https://doi.org/10.3233/ASY-221799 doi: 10.3233/ASY-221799
![]() |
[27] | Q. Wang, D. Zhao, Existence and mass concentration of 2D attractive Bose-Einstein condensates with periodic potentials, J. Differ. Equ., 262 (2017), 2684–2704. |
[28] |
J. Yang, H. Chen, L. Liu, Limiting behaviors of constrained minimizers for the mass subcritical fractional NLS equations, Anal. Math. Phys., 14 (2024), 32. https://doi.org/10.1007/s13324-024-00899-x doi: 10.1007/s13324-024-00899-x
![]() |
[29] |
S. Yao, H. Chen, V. Rădulescu, J. Sun, Normalized solutions for lower critical Choquard equations with critical Sobolev perturbations, SIAM J. Math. Anal., 54 (2022), 3696–3723. https://doi.org/10.1137/21M1463136 doi: 10.1137/21M1463136
![]() |