Existence and multiplicity of positive solutions to sub-elliptic systems with multiple critical exponents on Carnot groups

Abstract

In this work, we study the Nehari manifold and its application to the following sub-elliptic system involving the multiple critical Sobolev exponents:

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta _{\mathbb {G}}u=|u|^{2^*-2}u + \frac{\lambda \alpha }{\alpha +\beta }|u|^{\alpha -2}u |v|^{\beta } + \mu g(z)|u|^{q-2}u,&{}\quad z\in \Omega ,\\ -\Delta _{\mathbb {G}}v= |v|^{2^*-2}v + \frac{\lambda \beta }{\alpha +\beta }|u|^{\alpha }|v|^{\beta -2}v +\nu h(z)|v|^{q-2}v,&{}\quad z\in \Omega ,\\ u = v = 0, &{}\quad z\in \partial \Omega , \end{array}\right. \end{aligned}$$

where $\Omega$ is an open bounded subset of $\mathbb {G}$ with smooth boundary, $-\Delta _{\mathbb {G}}$ is the sub-Laplacian on Carnot group $\mathbb {G}$, $\lambda > 0$ is a parameter, $1< q < 2$, $\alpha$, $\beta >1$, $\alpha +\beta =2^*$, $2^*=\frac{2Q}{Q-2}$ is the critical Sobolev exponent, and Q is the homogeneous dimension of $\mathbb {G}$. By exploiting the Nehari manifold and variational methods, we prove that the system has at least two positive solutions when the pair of the parameters $(\mu ,\nu )$ belongs to a certain subset of $\mathbb {R}^{2}_{+}$.