A STRONGLY COUPLED SUB-LAPLACIAN SYSTEM ON THE HEISENBERG GROUP ℍ₁

Abstract

In this work, we study the following sub-elliptic system involving strongly coupledcritical terms and concave nonlinearities:

$$

\begin{cases}

-\Delta_{\mathbb{H}_{1}} u=\dfrac{\eta_1 \alpha_1}{2^*}\vert u\vert ^{\alpha_1-2}\vert v\vert ^{\beta_1} u+\dfrac{\eta_2 \alpha_2}{2^*}\vert u\vert ^{\alpha_2-2}\vert v\vert ^{\beta_2} u+\lambda g(z) \vert u\vert^{q-2} u, & z \in \Omega, \\

-\Delta_{\mathbb{H}_{1}}v=\dfrac{\eta_1 \beta_1}{2^*}\vert u\vert ^{\alpha_1}\vert v\vert ^{\beta_1-2} v+\dfrac{\eta_2 \beta_2}{2^*}\vert u\vert ^{\alpha_2}\vert v\vert ^{\beta_2-2} v+\mu h(z) \vert v\vert^{q-2} v, & z \in \Omega, \\

u=v=0, & z \in \partial\Omega,\end{cases}

$$

where is an open bounded subset of ℍ₁ with smooth boundary, -ΔH1 is the sub-Laplacian on Heisenberg group ℍ₁, η1, η2, λ, μ are positive, α1+β1 = 2*, α2+β2 = 2*, 1<q<2, 2={2Q}over{Q-2} is the critical Sobolev exponent on the Heisenberg group with Q=4 the homogeneous dimension of ℍ₁. By exploiting the Nehari manifold and variational methods, we prove that the system has at least two positive solutions.