# American Institute of Mathematical Sciences

• Previous Article
Global boundedness for a chemotaxis-competition system with signal dependent sensitivity and loop
• ERA Home
• This Issue
• Next Article
Enhancement of gamma oscillations in E/I neural networks by increase of difference between external inputs
November  2021, 29(5): 3243-3260. doi: 10.3934/era.2021036

## 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 Published  November 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^*_{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, 2021, 29 (5) : 3243-3260. doi: 10.3934/era.2021036 ##### 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. Barrios, E. Colorado, A. 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. Brändle, E. Colorado, A. 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 Nápoli, J. 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. Ferrari, M. Miranda Jr, D. Pallara, A. 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. Ghoussoub, F. Robert, S. 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. Zhang, X. 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 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. Barrios, E. Colorado, A. 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. Brändle, E. Colorado, A. 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 Nápoli, J. 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. Ferrari, M. Miranda Jr, D. Pallara, A. 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. Ghoussoub, F. Robert, S. 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. Zhang, X. 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  [1] Qingfang Wang. The Nehari manifold for a fractional Laplacian equation involving critical nonlinearities. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2261-2281. doi: 10.3934/cpaa.2018108 [2] Salvatore A. Marano, Nikolaos S. Papageorgiou. Positive solutions to a Dirichlet problem with$p$-Laplacian and concave-convex nonlinearity depending on a parameter. Communications on Pure & Applied Analysis, 2013, 12 (2) : 815-829. doi: 10.3934/cpaa.2013.12.815 [3] Xiaoming He, Marco Squassina, Wenming Zou. The Nehari manifold for fractional systems involving critical nonlinearities. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1285-1308. doi: 10.3934/cpaa.2016.15.1285 [4] M. L. M. Carvalho, Edcarlos D. Silva, C. Goulart. Choquard equations via nonlinear rayleigh quotient for concave-convex nonlinearities. Communications on Pure & Applied Analysis, 2021, 20 (10) : 3445-3479. doi: 10.3934/cpaa.2021113 [5] Jia-Feng Liao, Yang Pu, Xiao-Feng Ke, Chun-Lei Tang. Multiple positive solutions for Kirchhoff type problems involving concave-convex nonlinearities. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2157-2175. doi: 10.3934/cpaa.2017107 [6] Lucas C. F. Ferreira, Elder J. Villamizar-Roa. On the heat equation with concave-convex nonlinearity and initial data in weak-$L^p$spaces. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1715-1732. doi: 10.3934/cpaa.2011.10.1715 [7] Miao-Miao Li, Chun-Lei Tang. Multiple positive solutions for Schrödinger-Poisson system in$\mathbb{R}^{3}$involving concave-convex nonlinearities with critical exponent. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1587-1602. doi: 10.3934/cpaa.2017076 [8] Ali Maalaoui. A note on commutators of the fractional sub-Laplacian on Carnot groups. Communications on Pure & Applied Analysis, 2019, 18 (1) : 435-453. doi: 10.3934/cpaa.2019022 [9] Boumediene Abdellaoui, Abdelrazek Dieb, Enrico Valdinoci. A nonlocal concave-convex problem with nonlocal mixed boundary data. Communications on Pure & Applied Analysis, 2018, 17 (3) : 1103-1120. doi: 10.3934/cpaa.2018053 [10] Junping Shi, Ratnasingham Shivaji. Exact multiplicity of solutions for classes of semipositone problems with concave-convex nonlinearity. Discrete & Continuous Dynamical Systems, 2001, 7 (3) : 559-571. doi: 10.3934/dcds.2001.7.559 [11] Qingfang Wang. Multiple positive solutions of fractional elliptic equations involving concave and convex nonlinearities in$R^N$. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1671-1688. doi: 10.3934/cpaa.2016008 [12] Zaizheng Li, Qidi Zhang. Sub-solutions and a point-wise Hopf's lemma for fractional$ p $-Laplacian. Communications on Pure & Applied Analysis, 2021, 20 (2) : 835-865. doi: 10.3934/cpaa.2020293 [13] João Marcos do Ó, Uberlandio Severo. Quasilinear Schrödinger equations involving concave and convex nonlinearities. Communications on Pure & Applied Analysis, 2009, 8 (2) : 621-644. doi: 10.3934/cpaa.2009.8.621 [14] Caisheng Chen, Qing Yuan. Existence of solution to$p-$Kirchhoff type problem in$\mathbb{R}^N$via Nehari manifold. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2289-2303. doi: 10.3934/cpaa.2014.13.2289 [15] Sophia Th. Kyritsi, Nikolaos S. Papageorgiou. Positive solutions for p-Laplacian equations with concave terms. Conference Publications, 2011, 2011 (Special) : 922-930. doi: 10.3934/proc.2011.2011.922 [16] Yaoping Chen, Jianqing Chen. Existence of multiple positive weak solutions and estimates for extremal values for a class of concave-convex elliptic problems with an inverse-square potential. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1531-1552. doi: 10.3934/cpaa.2017073 [17] Hongyu Ye. Positive high energy solution for Kirchhoff equation in$\mathbb{R}^{3}$with superlinear nonlinearities via Nehari-Pohožaev manifold. Discrete & Continuous Dynamical Systems, 2015, 35 (8) : 3857-3877. doi: 10.3934/dcds.2015.35.3857 [18] Mingzheng Sun, Jiabao Su, Leiga Zhao. Infinitely many solutions for a Schrödinger-Poisson system with concave and convex nonlinearities. Discrete & Continuous Dynamical Systems, 2015, 35 (1) : 427-440. doi: 10.3934/dcds.2015.35.427 [19] Said Taarabti. Positive solutions for the$ p(x)- \$Laplacian : Application of the Nehari method. Discrete & Continuous Dynamical Systems - S, 2022, 15 (1) : 229-243. doi: 10.3934/dcdss.2021029 [20] Carlo Mercuri, Michel Willem. A global compactness result for the p-Laplacian involving critical nonlinearities. Discrete & Continuous Dynamical Systems, 2010, 28 (2) : 469-493. doi: 10.3934/dcds.2010.28.469

2020 Impact Factor: 1.833