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November  2021, 20(11): 3723-3744. doi: 10.3934/cpaa.2021128

Normalized solutions of supercritical nonlinear fractional Schrödinger equation with potential

School of Mathematics and Statistics, Jiangxi Normal University, Nanchang, Jiangxi 330022, China

* Corresponding author

Received  November 2020 Revised  June 2021 Published  November 2021 Early access  July 2021

Fund Project: The second author is supported by the National Natural Science Foundation of China (Grant Nos. 11701239 and 11871253)

We are concerned with the following nonlinear fractional Schrödinger equation:
$\begin{equation} (-\Delta)^s u+V(x)u+\omega u = |u|^{p-2}u\quad {\rm{in}}\,\,{\mathbb{R}}^N,\;\;\;\;\;\;({\textbf{P}})\end{equation}$
where
$ s\in(0,1) $
and
$ p\in\left(2+4s/N,2^*_s\right) $
, that is, the mass supercritical and Sobolev subcritical. Under certain assumptions on the potential
$ V:{\mathbb{R}}^N\rightarrow {\mathbb{R}} $
, positive and vanishing at infinity including potentials with singularities (which is important for physical reasons), we prove that there exists at least one
$ L^2 $
-normalized solution
$ (u,\omega)\in H^s({\mathbb{R}}^N)\times{\mathbb{R}}^+ $
of equation (P). In order to overcome the lack of compactness, the proof is based on a new min-max argument and splitting lemma for nonlocal version.
Citation: Songbai Peng, Aliang Xia. Normalized solutions of supercritical nonlinear fractional Schrödinger equation with potential. Communications on Pure and Applied Analysis, 2021, 20 (11) : 3723-3744. doi: 10.3934/cpaa.2021128
References:
[1]

T. BartschL. Jeanjean and N. Soave, Normalized solutions for a system of coupled cubic Schrödinger equations on ${\mathbb{R}}^3$, J. Math. Pures Appl., 9 (2016), 583-614.  doi: 10.1016/j.matpur.2016.03.004.

[2]

T. Bartsch, R. Molle, M. Rizzi and G. Verzini, Normalized solutions of mass supercritical Schrödinger equations with potential, Commun. Partial Differ. Equ., (2021), 28pp. doi: https://doi.org/10.1080/03605302.2021.1893747.

[3]

T. Bartsch and T. Weth, Three nodal solutions of singularly perturbed elliptic equations on domains without topology, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 259-281.  doi: 10.1016/j.anihpc.2004.07.005.

[4]

V. Benci and G. Cerami, Positive solutions of some nonlinear elliptic problems in exterior domains, Arch. Rational Mech. Anal., 99 (1987), 283-300.  doi: 10.1007/BF00282048.

[5]

D. BonheureJ. B. CasterasT. Gou and L. Jeanjean, Normalized solutions to the mixed dispersion nonlinear Schrödinger equation in the mass critical and supercritical regime, Trans. Amer. Math. Soc., 372 (2019), 2167-2212.  doi: 10.1090/tran/7769.

[6]

T. BoulengerD. Himmelsbach and E. Lenzmann, Blowup for fractional NLS, J. Funct. Anal., 271 (2016), 2569-2603.  doi: 10.1016/j.jfa.2016.08.011.

[7]

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.

[8]

J. A. Cardoso, D. S. dos Prazeres and U. B. Severo, Fractional Schrödinger equations involving potential vanishing at infinity and supercritical exponents, Z. Angew. Math. Phys., 71 (2020), 14pp. doi: 10.1007/s00033-020-01354-0.

[9]

G. Cerami and D. Passaseo, The effect of concentrating potentials in some singularly perturbed problems, Calc. Var. Partial Differ. Equ., 17 (2003), 257-281.  doi: 10.1007/s00526-002-0169-6.

[10]

M. Cheng, Bound state for the fractional Schrödinger equation with unbounded potential, J. Math. Phys., 53 (2012), 7pp. doi: 10.1063/1.3701574.

[11] R. Cont and P. Tankov, Financial modelling with jump processes, Chapman and Hall/CRC Financ. Math. Ser., Chapman and Hall/CRC press, 2004. 
[12]

J. Correia and G. Figueiredo, Existence of positive solution for a fractional elliptic equation in exterior domain, J. Differ. Equ., 268 (2020), 1946-1973.  doi: 10.1016/j.jde.2019.09.024.

[13]

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.

[14]

J. DávilaM. del Pino and J. Wei, Concentrating standing waves for the fractional nonlinear Schrödinger equation, J. Differ. Equ., 256 (2014), 858-892.  doi: 10.1016/j.jde.2013.10.006.

[15]

E. Di NezzaG. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional sobolev spaces, Bull. des Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.

[16]

M. DuL. TianJ. Wang and F. Zhang, Existence of normalized solutions for nonlinear fractional Schrödinger equations with trapping potentials, Proc. Roy. Soc. Edinburgh Sect. A, 149 (2019), 617-653.  doi: 10.1017/prm.2018.41.

[17]

P. FelmerA. Quaas and J. Tan, Positive solutions of nonlinear Schrödinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 142 (2012), 1237-1262.  doi: 10.1017/S0308210511000746.

[18]

B. Feng, J. Ren and Q. Wang, Existence and instability of normalized standing waves for the fractional Schrödinger equations in the $L^2$-supercritical case, J. Math. Phys., 61 (2020), 19pp. doi: 10.1063/5.0006247.

[19]

R. FrankE. Lenzmann and L. Silvestre, Uniqueness of radial solutions for the fractional Laplacian, Commun. Pure Appl. Math., 69 (2016), 1671-1726.  doi: 10.1002/cpa.21591.

[20] N. Ghoussoub, Duality and Perturbation Methods in Critical Point Theory, Cambridge University Press, Cambridge, 1993.  doi: 10.1017/CBO9780511551703.
[21]

Y. GuoZ. Q. WangX. Zeng and H. Zhou, Properties of ground states of attractive Gross-Pitaevskii equations with multi-well potentials, Nonlinearity, 31 (2018), 957-979.  doi: 10.1088/1361-6544/aa99a8.

[22]

X. He and W. Zou, Bifurcation and multiplicity of positive solutions for nonhomogeneous fractional Schrödinger equations with critical growth, Sci. China Math., 63 (2020), 1571-1612.  doi: 10.1007/s11425-020-1692-1.

[23]

X. He and W. Zou, Existence and concentration result for the fractional Schrödinger equations with critical nonlinearities, Calc. Var. Partial Differ. Equ., 55 (2016), 39pp. doi: 10.1007/s00526-016-1045-0.

[24]

N. Ikoma and Y. Miyamoto, Stable standing waves of nonlinear Schrödinger equations with potentials and general nonlinearities, Calc. Var. Partial Differ. Equ., 48 (2020), 20pp. doi: 10.1007/s00526-020-1703-0.

[25]

L. Jeanjean, Existence of solutions with prescribed norm for semilinear elliptic equations, Nonlinear Anal., 28 (1997), 1633-1659.  doi: 10.1016/S0362-546X(96)00021-1.

[26]

N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298-305.  doi: 10.1016/S0375-9601(00)00201-2.

[27]

N. Laskin, Fractional Schrödinger equation, Phys. Rev. E, 3 (2002), 7pp. doi: 10.1103/PhysRevE. 66.056108.

[28]

H. Luo and Z. Zhang, Normalized solutions to the fractional Schrödinger equations with combined nonlinearities, Calc. Var. Partial Differ. Equ., 59 (2020), 35pp. doi: 10.1007/s00526-020-01814-5.

[29]

S. Secchi, Ground state solutions for nonlinear fractional Schrödinger equations in ${\mathbb{R}}^N$, J. Math. Phys., 54 (2013), 17pp. doi: 10.1063/1.4793990.

[30]

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.

[31]

X. Shang and J. Zhang, Multiplicity and concentration of positive solutions for fractional nonlinear Schrödinger equation, Commun. Pure Appl. Anal., 17 (2018), 2239-2259.  doi: 10.3934/cpaa.2018107.

[32]

N. Soave, Normalized ground states for the NLS equation with combined nonlinearities: the Sobolev critical case, J. Funct. Anal., 279 (2020), 43pp. doi: 10.1016/j. jfa. 2020.108610.

[33]

A. Xia and J. Yang, Normalized solutions of higher-order Schrödinger equations, Discrete Contin. Dyn. Syst., 39 (2019), 447-462.  doi: 10.3934/dcds.2019018.

[34]

J. Yang and J. Yang, On supercritical nonlinear Schrödinger equations with ellipse-shaped potentials, Proc. Roy. Soc. Edinburgh Sect. A, 150 (2020), 3187-3215.  doi: 10.1017/prm.2019.66.

show all references

References:
[1]

T. BartschL. Jeanjean and N. Soave, Normalized solutions for a system of coupled cubic Schrödinger equations on ${\mathbb{R}}^3$, J. Math. Pures Appl., 9 (2016), 583-614.  doi: 10.1016/j.matpur.2016.03.004.

[2]

T. Bartsch, R. Molle, M. Rizzi and G. Verzini, Normalized solutions of mass supercritical Schrödinger equations with potential, Commun. Partial Differ. Equ., (2021), 28pp. doi: https://doi.org/10.1080/03605302.2021.1893747.

[3]

T. Bartsch and T. Weth, Three nodal solutions of singularly perturbed elliptic equations on domains without topology, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 259-281.  doi: 10.1016/j.anihpc.2004.07.005.

[4]

V. Benci and G. Cerami, Positive solutions of some nonlinear elliptic problems in exterior domains, Arch. Rational Mech. Anal., 99 (1987), 283-300.  doi: 10.1007/BF00282048.

[5]

D. BonheureJ. B. CasterasT. Gou and L. Jeanjean, Normalized solutions to the mixed dispersion nonlinear Schrödinger equation in the mass critical and supercritical regime, Trans. Amer. Math. Soc., 372 (2019), 2167-2212.  doi: 10.1090/tran/7769.

[6]

T. BoulengerD. Himmelsbach and E. Lenzmann, Blowup for fractional NLS, J. Funct. Anal., 271 (2016), 2569-2603.  doi: 10.1016/j.jfa.2016.08.011.

[7]

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.

[8]

J. A. Cardoso, D. S. dos Prazeres and U. B. Severo, Fractional Schrödinger equations involving potential vanishing at infinity and supercritical exponents, Z. Angew. Math. Phys., 71 (2020), 14pp. doi: 10.1007/s00033-020-01354-0.

[9]

G. Cerami and D. Passaseo, The effect of concentrating potentials in some singularly perturbed problems, Calc. Var. Partial Differ. Equ., 17 (2003), 257-281.  doi: 10.1007/s00526-002-0169-6.

[10]

M. Cheng, Bound state for the fractional Schrödinger equation with unbounded potential, J. Math. Phys., 53 (2012), 7pp. doi: 10.1063/1.3701574.

[11] R. Cont and P. Tankov, Financial modelling with jump processes, Chapman and Hall/CRC Financ. Math. Ser., Chapman and Hall/CRC press, 2004. 
[12]

J. Correia and G. Figueiredo, Existence of positive solution for a fractional elliptic equation in exterior domain, J. Differ. Equ., 268 (2020), 1946-1973.  doi: 10.1016/j.jde.2019.09.024.

[13]

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.

[14]

J. DávilaM. del Pino and J. Wei, Concentrating standing waves for the fractional nonlinear Schrödinger equation, J. Differ. Equ., 256 (2014), 858-892.  doi: 10.1016/j.jde.2013.10.006.

[15]

E. Di NezzaG. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional sobolev spaces, Bull. des Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.

[16]

M. DuL. TianJ. Wang and F. Zhang, Existence of normalized solutions for nonlinear fractional Schrödinger equations with trapping potentials, Proc. Roy. Soc. Edinburgh Sect. A, 149 (2019), 617-653.  doi: 10.1017/prm.2018.41.

[17]

P. FelmerA. Quaas and J. Tan, Positive solutions of nonlinear Schrödinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 142 (2012), 1237-1262.  doi: 10.1017/S0308210511000746.

[18]

B. Feng, J. Ren and Q. Wang, Existence and instability of normalized standing waves for the fractional Schrödinger equations in the $L^2$-supercritical case, J. Math. Phys., 61 (2020), 19pp. doi: 10.1063/5.0006247.

[19]

R. FrankE. Lenzmann and L. Silvestre, Uniqueness of radial solutions for the fractional Laplacian, Commun. Pure Appl. Math., 69 (2016), 1671-1726.  doi: 10.1002/cpa.21591.

[20] N. Ghoussoub, Duality and Perturbation Methods in Critical Point Theory, Cambridge University Press, Cambridge, 1993.  doi: 10.1017/CBO9780511551703.
[21]

Y. GuoZ. Q. WangX. Zeng and H. Zhou, Properties of ground states of attractive Gross-Pitaevskii equations with multi-well potentials, Nonlinearity, 31 (2018), 957-979.  doi: 10.1088/1361-6544/aa99a8.

[22]

X. He and W. Zou, Bifurcation and multiplicity of positive solutions for nonhomogeneous fractional Schrödinger equations with critical growth, Sci. China Math., 63 (2020), 1571-1612.  doi: 10.1007/s11425-020-1692-1.

[23]

X. He and W. Zou, Existence and concentration result for the fractional Schrödinger equations with critical nonlinearities, Calc. Var. Partial Differ. Equ., 55 (2016), 39pp. doi: 10.1007/s00526-016-1045-0.

[24]

N. Ikoma and Y. Miyamoto, Stable standing waves of nonlinear Schrödinger equations with potentials and general nonlinearities, Calc. Var. Partial Differ. Equ., 48 (2020), 20pp. doi: 10.1007/s00526-020-1703-0.

[25]

L. Jeanjean, Existence of solutions with prescribed norm for semilinear elliptic equations, Nonlinear Anal., 28 (1997), 1633-1659.  doi: 10.1016/S0362-546X(96)00021-1.

[26]

N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298-305.  doi: 10.1016/S0375-9601(00)00201-2.

[27]

N. Laskin, Fractional Schrödinger equation, Phys. Rev. E, 3 (2002), 7pp. doi: 10.1103/PhysRevE. 66.056108.

[28]

H. Luo and Z. Zhang, Normalized solutions to the fractional Schrödinger equations with combined nonlinearities, Calc. Var. Partial Differ. Equ., 59 (2020), 35pp. doi: 10.1007/s00526-020-01814-5.

[29]

S. Secchi, Ground state solutions for nonlinear fractional Schrödinger equations in ${\mathbb{R}}^N$, J. Math. Phys., 54 (2013), 17pp. doi: 10.1063/1.4793990.

[30]

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.

[31]

X. Shang and J. Zhang, Multiplicity and concentration of positive solutions for fractional nonlinear Schrödinger equation, Commun. Pure Appl. Anal., 17 (2018), 2239-2259.  doi: 10.3934/cpaa.2018107.

[32]

N. Soave, Normalized ground states for the NLS equation with combined nonlinearities: the Sobolev critical case, J. Funct. Anal., 279 (2020), 43pp. doi: 10.1016/j. jfa. 2020.108610.

[33]

A. Xia and J. Yang, Normalized solutions of higher-order Schrödinger equations, Discrete Contin. Dyn. Syst., 39 (2019), 447-462.  doi: 10.3934/dcds.2019018.

[34]

J. Yang and J. Yang, On supercritical nonlinear Schrödinger equations with ellipse-shaped potentials, Proc. Roy. Soc. Edinburgh Sect. A, 150 (2020), 3187-3215.  doi: 10.1017/prm.2019.66.

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