On the existence and regularity of vector solutions for quasilinear systems with linear coupling

We study the following coupled system of quasilinear equations: $$\begin{cases}-\Delta_pu+|u|^{p-2}u=f(u)+\lambda v, & x \in \mathbb{R}^N,\\-\Delta_pv+|v|^{p-2}v=g(v)+\lambda u, & x \in \mathbb{R}^N.\end{cases}$${−Δpu+|u|p−2u=f(u)+λv,x∈RN,−Δpv+|v|p−2v=g(v)+λu,x∈RN..Under some assumptions on the nonlinear terms f and g, we establish some results about the existence and regularity of vector solutions for the p-Laplacian systems by using variational methods. In particular, we get two pairs of nontrivial solutions. We also study the different asymptotic behavior of solutions as the coupling parameter λ tends to zero.