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

We prove existence and regularity results for the following elliptic system: $$\{\begin{array}{ll}-\mathbf{\text{div}}(\mathrm{\mid }D\mathit{u}{\mathrm{\mid }}^{p-2}D\mathit{u})=\mathit{f}(x,\mathit{u})& \text{in }\mathrm{\Omega }\\ \mathit{u}=0& \text{on }\mathrm{\partial }\mathrm{\Omega },\end{array}$$ where $\mathit{u}=(u^1,\dots ,u^m)$, $p>1$, and $\mathrm{\Omega }\subset {\mathbb{R}}^N$ is a bounded domain. We also consider the special case $$\mathit{f}(x,\mathit{u})=\lambda \mathrm{\mid }\mathit{u}{\mathrm{\mid }}^{p-2}\mathit{u}+\mathrm{\mid }\mathit{u}{\mathrm{\mid }}^{q-2}\mathit{u},$$ 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 $\mathit{c}=(c^1,\dots ,c^m)\in S^{m-1}$ (the $(m-1)$-sphere in ${\mathbb{R}}^m$) and $\omega$ is a positive solution of the corresponding scalar equation.