### Electronic Research Archive

2021, Issue 5: 3243-3260. doi: 10.3934/era.2021036

# Fractional $p$-sub-Laplacian operator problem with concave-convex nonlinearities on homogeneous groups

• Received: 01 November 2020 Revised: 01 March 2021 Published: 26 May 2021
• Primary: 22E30, 35H20; Secondary: 43A80

• 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[J]. Electronic Research Archive, 2021, 29(5): 3243-3260. doi: 10.3934/era.2021036

### Related Papers:

• 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 $(0,+\infty)$.

 [1] Approximations of Sobolev norms in Carnot groups. Commun. Contemp. Math. (2011) 13: 765-794. [2] On some critical problems for the fractional Laplacian operator. J. Differential Equations (2012) 252: 6133-6162. [3] A. Bonfiglioli, E. Lanconelli and F. Uguzzoni, Stratified Lie Groups and Potential Theory for Their Sub-Laplacians, Springer, 2007. [4] A concave-convex elliptic problem involving the fractional Laplacian. Proc. Roy. Soc. Edinburgh Sect. A (2013) 143: 39-71. [5] F. Buseghin, N. Garofalo and G. Tralli, On the limiting behavior of some nonlocal semi-norms: A new phenomenon, preprint (2020). [6] Positive solutions of nonlinear problems involving the square root of the Laplacian. Adv. Math. (2010) 224: 2052-2093. [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 [8] A Pólya-Szegö principle for general fractional Orlicz–Sobolev spaces. Complex Variables and Elliptic Equations (2020) 66: 1-23. [9] Fractional Laplacians, perimeters and heat semigroups in Carnot groups. Discrete Contin. Dyn. Syst. (Series S) (2018) 11: 477-491. [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 [11] (1982) Hardy Spaces on Homogeneous Groups.volume 28 of Mathematical Notes. Princeton University Press. [12] Fractional $p$-eigenvalues. Riv. Mat. Univ. Parma (2014) 5: 373-386. [13] Mass and asymptotics associated to fractional Hardy-Schrödinger operators in critical regimes. Comm Partial Differential Equations (2018) 43: 859-892. [14] Borderline variational problems involving fractional Laplacians and critical singularities. Advanced Nonlinear Studies (2015) 15: 527-555. [15] Nehari manifold for non-local elliptic operator with concave–convex nonlinearities and sign-changing weight functions. Proc. Indian Acad. Sci. Math. Sci. (2015) 125: 545-558. [16] Lyapunov-type inequalities for the fractional p-sub-Laplacian. Advances in Operator Theory (2020) 5: 435-452. [17] E. Lindgren and P. Lindqvist, Fractional eigenvalues, Calc. Var. Partial Differential Equations, 49 (2014), 795–-826. doi: 10.1007/s00526-013-0600-1 [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 [19] Mountain pass solutions for non-local elliptic operators. J. Math. Anal. Appl. (2012) 389: 887-898. [20] Variational methods for non-local operators of elliptic type. Discrete Contin. Dyn. Syst. (2013) 33: 2105-2137. [21] Regularity of the obstacle problem for a fractional power of the Laplace operator. Comm. Pure Appl. Math. (2007) 60: 67-112. [22] Three solutions for a fractional elliptic problems with critical and supercritical growth. Acta Mathematica Scientia (2016) 36: 1819-1831. [23] Multiplicity of positive solutions for a fractional laplacian equations involving critical nonlinearity. Topol. Methods Nonlinear Anal. (2019) 53: 151-182. [24] Multiplicity of positive solutions for a nonlocal elliptic problem involving critical Sobolev-Hardy exponents and concave-convex nonlinearities. Acta Math. Sci. (2020) 40B: 679-699. [25] Nonlocal elliptic systems involving critical Sobolev-Hardy exponents and concave-convex nonlinearities. Taiwanese J Math. (2019) 23: 1479-1510. [26] Multiple solutions for a fractional Laplacian system involving critical Sobolev-Hardy exponents and homogeneous term. Math. Mode. Anal. (2020) 25: 1-20. [27] Existence results for a fractional elliptic system with critical Sobolev-Hardy exponents and concave-convex nonlinearities. Math Meth Appl Sci. (2020) 43: 3488-3512.
###### 通讯作者: 陈斌, bchen63@163.com
• 1.

沈阳化工大学材料科学与工程学院 沈阳 110142

1.604 0.8

Article outline

Figures(7)

• On This Site