Existence results for variational quasilinear elliptic systems involving the vectorial $p$-Laplacian

We prove existence and regularity results for the following elliptic system: $$\begin{cases} -\textbf{div}(|D\boldsymbol{u}|^{p-2}D\boldsymbol{u})=\boldsymbol{f}(x,\boldsymbol{u}) & \text{in } \Omega \\ \boldsymbol{u}=0 & \text{on } \partial\Omega, \end{cases}$$ where $\boldsymbol{u}=(u^1,\dots,u^m)$, $p>1$, and $\Omega\subset\mathbb{R}^N$ is a bounded domain. We also consider the special case $$\boldsymbol{f}(x,\boldsymbol{u})=\lambda|\boldsymbol{u}|^{p-2}\boldsymbol{u}+|\boldsymbol{u}|^{q-2}\boldsymbol{u},$$ and we prove a classification result. In particular, we show that any least energy solution is of the form $(c^1\omega,\dots,c^m\omega)$, where $\boldsymbol{c}=(c^1,\dots,c^m)\in S^{m-1}$ (the $(m-1)$-sphere in $\mathbb R^m$) and $\omega$ is a positive solution of the corresponding scalar equation.