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

Abstract

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

Keywords: Subcritical nonlinearities, vectorial $p$-Laplacian, least energy solutions, Dirichlet boundary conditions, quasilinear elliptic systems, Lane-Emden equations