Existence results for variational quasilinear elliptic systems involving the vectorial $p$-Laplacian
Annamaria Canino, Simone Mauro
TL;DR
This work addresses the existence and regularity of solutions to a vector-valued quasilinear elliptic system driven by the vectorial $p$-Laplacian, under Dirichlet boundary conditions on a bounded domain. By formulating an energy functional $\mathcal J$ with subcritical growth and applying variational methods, the authors establish global minimizers in the $p$-sublinear case and obtain infinitely many weak solutions in the $p$-superlinear regime, using a PS framework and spectral decomposition. A notable contribution is the classification of least energy solutions for a Lane-Emden type nonlinearity, proving that minimizers must have the semitrivial form $\boldsymbol u=\boldsymbol c\,\omega$ with $\boldsymbol c\in S^{m-1}$ and $\omega$ solving a scalar $p$-Laplacian equation. The results combine regularity theory to upgrade weak solutions to $L^\infty$ and, under smooth boundaries, to $C^{1,\beta}$ regularity, offering a detailed picture of the interplay between nonlinear diffusion and subcritical reactions in multi-component systems.
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 $\boldsymbol{u}=(u^1,\dots,u^m)$, $p>1$, and $Ω\subset\mathbb{R}^N$ 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 $(c^1ω,\dots,c^mω)$, where $\boldsymbol{c}=(c^1,\dots,c^m)\in S^{m-1}$ (the $(m-1)$-sphere in $\mathbb R^m$) and $ω$ is a positive solution of the corresponding scalar equation.