A sub-elliptic system with strongly coupled critical terms and concave nonlinearities

Abstract

In this work, we study the Nehari manifold and its application to the following sub-elliptic system involving strongly coupled critical terms and concave nonlinearities:

\[
\left\{
\begin{aligned}
-\Delta_{\mathbb{G}} u
&= \frac{\eta_{1}\alpha_{1}}{2^*}\,|u|^{\alpha_{1}-2}|v|^{\beta_{1}}u
+\frac{\eta_{2}\alpha_{2}}{2^*}\,|u|^{\alpha_{2}-2}|v|^{\beta_{2}}u
\\ &+\lambda\, g(z)\,|u|^{q-2}u,
&& z\in\Omega, \\
-\Delta_{\mathbb{G}} v
&= \frac{\eta_{1}\beta_{1}}{2^*}\,|u|^{\alpha_{1}}|v|^{\beta_{1}-2}v
+\frac{\eta_{2}\beta_{2}}{2^*}\,|u|^{\alpha_{2}}|v|^{\beta_{2}-2}v
\\ &+\mu\, h(z)\,|v|^{q-2}v,
&& z\in\Omega, \\
u &= v = 0, && z\in\partial\Omega,
\end{aligned}
\right.
\]

where \(\Omega\) is an open bounded subset of \(\mathbb{G}\) with smooth boundary, \(-\Delta_{\mathbb{G}}\) is the sub-Laplacian on a Carnot group \(\mathbb{G}\); \(\eta_1, \eta_2, \lambda, \mu\) are positive, \(\alpha_1+\beta_1=2^*\), \(\alpha_2+\beta_2=2^*\), \(1<q<2\), \(2^*=\frac{2Q}{Q-2}\) 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.