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

We prove existence and regularity results for the following elliptic system: \[ {cases} -div(|Du|ᵖ⁻²Du)=f(x,u) & in Ω\\ u=0 & on ∂Ω, {cases} \] where u=(u¹,,uᵐ), $p>1$, and ΩN is a bounded domain. We also consider the special case \[f(x,u)=λ|u|ᵖ⁻²u+|u|q-2u,\] and we prove a classification result. In particular, we show that any least energy solution is of the form (c¹ω,,cᵐω), where c=(c¹,,cᵐ)∈ Sᵐ⁻¹ (the $(m-1)$-sphere in Rᵐ) and $ω$ is a positive solution of the corresponding scalar equation.