doi: 10.3934/era.2021036
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Fractional $ p $-sub-Laplacian operator problem with concave-convex nonlinearities on homogeneous groups

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

* Corresponding author: Jinguo Zhang

Received  November 2020 Revised  March 2021 Early access May 2021

Fund Project: This work was supported by the National Natural Science Foundation of China (Grant No.11761049)

This study examines the existence and multiplicity of non-negative solutions of the following fractional
$ p $
-sub-Laplacian problem
$ \begin{equation*} \left\{\begin{aligned} &(-\Delta_{p,g})^{s}u = \lambda f(x)|u|^{\alpha-2}u+ h(x)|u|^{\beta-2} u \quad&\rm{in}\,\,\, &\Omega,\\ &\,\,\, u = 0\quad\quad &\rm{in} \,\,\, &\mathbb{G}\setminus \Omega, \end{aligned}\right. \end{equation*} $
where
$ \Omega $
is an open bounded in homogeneous Lie group
$ \mathbb{G} $
with smooth boundary,
$ p>1 $
,
$ s\in(0,1) $
,
$ (-\Delta_{p,g})^{s} $
is the fractional
$ p $
-sub-Laplacian operator with respect to the quasi-norm
$ g $
,
$ \lambda>0 $
,
$ 1< \alpha<p <\beta < p^*_{s} $
,
$ p^*_{s}: = \frac{Qp}{Q-sp} $
is the fractional critical Sobolev exponents,
$ Q $
is the homogeneous dimensions of the homogeneous Lie group
$ \mathbb{G} $
with
$ Q> sp $
, and
$ f $
,
$ h $
are sign-changing smooth functions. With the help of the Nehari manifold, we prove that the nonlocal problem on homogeneous group has at least two nontrivial solutions when the parameter
$ \lambda $
belong to a center subset of
$ (0,+\infty) $
.
Citation: Jinguo Zhang, Dengyun Yang. Fractional $ p $-sub-Laplacian operator problem with concave-convex nonlinearities on homogeneous groups. Electronic Research Archive, doi: 10.3934/era.2021036
References:
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D. Barbieri, Approximations of Sobolev norms in Carnot groups, Commun. Contemp. Math., 13 (2011), 765-794.  doi: 10.1142/S0219199711004439.  Google Scholar

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B. BarriosE. ColoradoA. de Pablo and U. Sánchez, On some critical problems for the fractional Laplacian operator, J. Differential Equations, 252 (2012), 6133-6162.  doi: 10.1016/j.jde.2012.02.023.  Google Scholar

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C. BrändleE. ColoradoA. de Pablo and U. Sánchez, A concave-convex elliptic problem involving the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 143 (2013), 39-71.  doi: 10.1017/S0308210511000175.  Google Scholar

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F. Buseghin, N. Garofalo and G. Tralli, On the limiting behavior of some nonlocal semi-norms: A new phenomenon, preprint (2020). Google Scholar

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X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. Math., 224 (2010), 2052-2093.  doi: 10.1016/j.aim.2010.01.025.  Google Scholar

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P. De NápoliJ. Fernández Bonder and A. Salort, A Pólya-Szegö principle for general fractional Orlicz–Sobolev spaces, Complex Variables and Elliptic Equations, 66 (2020), 1-23.  doi: 10.1080/17476933.2020.1729139.  Google Scholar

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G. Franzina and G. Palatucci, Fractional $p$-eigenvalues, Riv. Mat. Univ. Parma, 5 (2014), 373-386.   Google Scholar

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N. GhoussoubF. RobertS. Shakerian and M. Zhao, Mass and asymptotics associated to fractional Hardy-Schrödinger operators in critical regimes, Comm Partial Differential Equations, 43 (2018), 859-892.  doi: 10.1080/03605302.2018.1476528.  Google Scholar

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S. Goyal and K. Sreenadh, Nehari manifold for non-local elliptic operator with concave–convex nonlinearities and sign-changing weight functions, Proc. Indian Acad. Sci. Math. Sci., 125 (2015), 545-558.  doi: 10.1007/s12044-015-0244-5.  Google Scholar

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A. Kassymov and D. Suragan, Lyapunov-type inequalities for the fractional p-sub-Laplacian, Advances in Operator Theory, 5 (2020), 435-452.  doi: 10.1007/s43036-019-00037-6.  Google Scholar

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M. Ruzhansky, N. Tokmagambetov and N. Yessirkegenov, Best constants in Sobolev and Gagliardo-Nirenberg inequalities on graded groups and ground states for higher order nonlinear subelliptic equations, Calc. Var. Partial Differential Equations, 59 (2020), Paper No. 175, 23 pp. doi: 10.1007/s00526-020-01835-0.  Google Scholar

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L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112.  doi: 10.1002/cpa.20153.  Google Scholar

[22]

J. Zhang and X. Liu, Three solutions for a fractional elliptic problems with critical and supercritical growth, Acta Mathematica Scientia, 36 (2016), 1819-1831.  doi: 10.1016/S0252-9602(16)30108-4.  Google Scholar

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J. ZhangX. Liu and H. Jiao, Multiplicity of positive solutions for a fractional laplacian equations involving critical nonlinearity, Topol. Methods Nonlinear Anal., 53 (2019), 151-182.  doi: 10.12775/tmna.2018.043.  Google Scholar

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J. Zhang and T.-S. Hsu, Multiplicity of positive solutions for a nonlocal elliptic problem involving critical Sobolev-Hardy exponents and concave-convex nonlinearities, Acta Math. Sci., 40B (2020), 679-699.  doi: 10.1007/s10473-020-0307-2.  Google Scholar

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J. Zhang and T.-S. Hsu, Nonlocal elliptic systems involving critical Sobolev-Hardy exponents and concave-convex nonlinearities, Taiwanese J Math., 23 (2019), 1479-1510.  doi: 10.11650/tjm/190109.  Google Scholar

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J. Zhang and T.-S. Hsu, Multiple solutions for a fractional Laplacian system involving critical Sobolev-Hardy exponents and homogeneous term, Math. Mode. Anal., 25 (2020), 1-20.  doi: 10.3846/mma.2020.7704.  Google Scholar

[27]

J. Zhang and T.-S. Hsu, Existence results for a fractional elliptic system with critical Sobolev-Hardy exponents and concave-convex nonlinearities, Math Meth Appl Sci., 43 (2020), 3488-3512.  doi: 10.1002/mma.6134.  Google Scholar

show all references

References:
[1]

D. Barbieri, Approximations of Sobolev norms in Carnot groups, Commun. Contemp. Math., 13 (2011), 765-794.  doi: 10.1142/S0219199711004439.  Google Scholar

[2]

B. BarriosE. ColoradoA. de Pablo and U. Sánchez, On some critical problems for the fractional Laplacian operator, J. Differential Equations, 252 (2012), 6133-6162.  doi: 10.1016/j.jde.2012.02.023.  Google Scholar

[3]

A. Bonfiglioli, E. Lanconelli and F. Uguzzoni, Stratified Lie Groups and Potential Theory for Their Sub-Laplacians, Springer, 2007.  Google Scholar

[4]

C. BrändleE. ColoradoA. de Pablo and U. Sánchez, A concave-convex elliptic problem involving the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 143 (2013), 39-71.  doi: 10.1017/S0308210511000175.  Google Scholar

[5]

F. Buseghin, N. Garofalo and G. Tralli, On the limiting behavior of some nonlocal semi-norms: A new phenomenon, preprint (2020). Google Scholar

[6]

X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. Math., 224 (2010), 2052-2093.  doi: 10.1016/j.aim.2010.01.025.  Google Scholar

[7]

M. Capolli, A. Maione, A. M. Salort and E. Vecchi, Asymptotic behaviours in fractional Orlicz-Sobolev spaces on Carnot groups, J. Geom. Anal., 31 (2020), 3196–-3229.. doi: 10.1007/s12220-020-00391-5.  Google Scholar

[8]

P. De NápoliJ. Fernández Bonder and A. Salort, A Pólya-Szegö principle for general fractional Orlicz–Sobolev spaces, Complex Variables and Elliptic Equations, 66 (2020), 1-23.  doi: 10.1080/17476933.2020.1729139.  Google Scholar

[9]

F. FerrariM. Miranda JrD. PallaraA. Pinamonti and Y. Sire, Fractional Laplacians, perimeters and heat semigroups in Carnot groups, Discrete Contin. Dyn. Syst. (Series S), 11 (2018), 477-491.  doi: 10.3934/dcdss.2018026.  Google Scholar

[10]

V. Fischer and M. Ruzhansky, Quantization on Nilpotent Lie Groups, volume 314 of Progress in Mathematics, Birkhäuser. (open access book), 2016 doi: 10.1007/978-3-319-29558-9.  Google Scholar

[11] G. B. Folland and E. M. Stein, Hardy Spaces on Homogeneous Groups, volume 28 of Mathematical Notes. Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1982.   Google Scholar
[12]

G. Franzina and G. Palatucci, Fractional $p$-eigenvalues, Riv. Mat. Univ. Parma, 5 (2014), 373-386.   Google Scholar

[13]

N. GhoussoubF. RobertS. Shakerian and M. Zhao, Mass and asymptotics associated to fractional Hardy-Schrödinger operators in critical regimes, Comm Partial Differential Equations, 43 (2018), 859-892.  doi: 10.1080/03605302.2018.1476528.  Google Scholar

[14]

N. Ghoussoub and S. Shakerian, Borderline variational problems involving fractional Laplacians and critical singularities, Advanced Nonlinear Studies, 15 (2015), 527-555.  doi: 10.1515/ans-2015-0302.  Google Scholar

[15]

S. Goyal and K. Sreenadh, Nehari manifold for non-local elliptic operator with concave–convex nonlinearities and sign-changing weight functions, Proc. Indian Acad. Sci. Math. Sci., 125 (2015), 545-558.  doi: 10.1007/s12044-015-0244-5.  Google Scholar

[16]

A. Kassymov and D. Suragan, Lyapunov-type inequalities for the fractional p-sub-Laplacian, Advances in Operator Theory, 5 (2020), 435-452.  doi: 10.1007/s43036-019-00037-6.  Google Scholar

[17]

E. Lindgren and P. Lindqvist, Fractional eigenvalues, Calc. Var. Partial Differential Equations, 49 (2014), 795–-826. doi: 10.1007/s00526-013-0600-1.  Google Scholar

[18]

M. Ruzhansky, N. Tokmagambetov and N. Yessirkegenov, Best constants in Sobolev and Gagliardo-Nirenberg inequalities on graded groups and ground states for higher order nonlinear subelliptic equations, Calc. Var. Partial Differential Equations, 59 (2020), Paper No. 175, 23 pp. doi: 10.1007/s00526-020-01835-0.  Google Scholar

[19]

R. Servadei and E. Valdinoci, Mountain pass solutions for non-local elliptic operators, J. Math. Anal. Appl., 389 (2012), 887-898.  doi: 10.1016/j.jmaa.2011.12.032.  Google Scholar

[20]

R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137.  doi: 10.3934/dcds.2013.33.2105.  Google Scholar

[21]

L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112.  doi: 10.1002/cpa.20153.  Google Scholar

[22]

J. Zhang and X. Liu, Three solutions for a fractional elliptic problems with critical and supercritical growth, Acta Mathematica Scientia, 36 (2016), 1819-1831.  doi: 10.1016/S0252-9602(16)30108-4.  Google Scholar

[23]

J. ZhangX. Liu and H. Jiao, Multiplicity of positive solutions for a fractional laplacian equations involving critical nonlinearity, Topol. Methods Nonlinear Anal., 53 (2019), 151-182.  doi: 10.12775/tmna.2018.043.  Google Scholar

[24]

J. Zhang and T.-S. Hsu, Multiplicity of positive solutions for a nonlocal elliptic problem involving critical Sobolev-Hardy exponents and concave-convex nonlinearities, Acta Math. Sci., 40B (2020), 679-699.  doi: 10.1007/s10473-020-0307-2.  Google Scholar

[25]

J. Zhang and T.-S. Hsu, Nonlocal elliptic systems involving critical Sobolev-Hardy exponents and concave-convex nonlinearities, Taiwanese J Math., 23 (2019), 1479-1510.  doi: 10.11650/tjm/190109.  Google Scholar

[26]

J. Zhang and T.-S. Hsu, Multiple solutions for a fractional Laplacian system involving critical Sobolev-Hardy exponents and homogeneous term, Math. Mode. Anal., 25 (2020), 1-20.  doi: 10.3846/mma.2020.7704.  Google Scholar

[27]

J. Zhang and T.-S. Hsu, Existence results for a fractional elliptic system with critical Sobolev-Hardy exponents and concave-convex nonlinearities, Math Meth Appl Sci., 43 (2020), 3488-3512.  doi: 10.1002/mma.6134.  Google Scholar

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