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

Jinguo Zhang; Dengyun Yang; Jinguo Zhang; Dengyun Yang

doi:10.3934/era.2021036

Electronic Research Archive

2021, Volume 29, Issue 5: 3243-3260. doi: 10.3934/era.2021036

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

Jinguo Zhang ^,,
Dengyun Yang

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

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) $.
- Fractional p-sub-Laplacian,
- concave-convex nonlinearities,
- Nehari manifold,
- homogeneous groups
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

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Abstract

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) $.

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