Jinguo Zhang, Sub-elliptic systems involving critical Hardy-Sobolev exponents and sign-changing weight functions on Carnot groups

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DOI: 10.23952/jnva.8.2024.2.02

Volume 8, Issue 2, 1 April 2024, Pages 199-231

Abstract. This paper concerns the existence and multiplicity of positive solutions for the following subelliptic singular system on Carnot group:

$-\Delta_{\mathbb{G}}u =\frac{p_{1}}{p_{1}+p_{2}} h(\xi) \frac{\psi^{\alpha}|u|^{p_{1}-2}u |v|^{p_{2}}}{d(\xi)^{\alpha}}+ \lambda f(\xi) \frac{\psi^{\beta}|u|^{q-2}u}{d(\xi)^{\beta}}$ in $\Omega,$

$-\Delta_{\mathbb{G}}v=\frac{p_{2}}{p_{1}+p_{2}}h(\xi) \frac{\psi^{\alpha}|u|^{p_{1}} |v|^{p_{2}-2} v}{d(\xi)^{\alpha}}+ \mu g(\xi) \frac{\psi^{\beta}|v|^{q-2}v}{d(\xi)^{\beta}}$ in $\Omega,$

$u=v=0$ on $\partial\Omega,$

where $-\Delta_{\mathbb{G}}$ is a sub-Laplacian on an arbitrary Carnot group $\mathbb{G}$ , $0\in \Omega$ , $d$ is the $\Delta_{\mathbb{G}}$ -gauge, $\psi=|\nabla_{\mathbb{G}}d|$ , $\Omega$ is a bounded domain in $\mathbb{G}$ with smooth boundary $\partial \Omega$ , $\lambda$ , $\mu>0$ , $1 \textless q \textless 2$ , $0\leq \alpha \textless 2$ , $0\leq \beta \textless 2$ , $p_{1}$ , $p_{2}>1$ satisfying $2 \textless p_{1}+p_{2}\leq 2^{*}(\alpha)$ with $2^{*}(\alpha)=\frac{2(Q-\alpha)}{Q-2}$ as a critical Hardy-Sobolev exponent in the Stratified Lie context. For suitable assumptions on weight functions $f(\xi)$ , $g(\xi)$ , and $h(\xi)$ , by using the variational methods and Nehari manifold, we prove that the subelliptic system admits at least two positive solutions when parameters pair $(\lambda, \mu)$ belongs to a certain subset of $\mathbb R^2_{+}$ .

How to Cite this Article:
J. Zhang, Sub-elliptic systems involving critical Hardy-Sobolev exponents and sign-changing weight functions on Carnot groups, J. Nonlinear Var. Anal. 8 (2024), 199-231.