On the uniqueness of eigenfunctions for the vectorial \(p\)-Laplacian (Q6139365)

The authors provide a short proof of the following fact first established in [\textit{F. Brock} and \textit{R. Manásevich}, Differ. Integral Equ. 20, No. 4, 429--444 (2007; Zbl 1212.35098)] for \(d \geq 1\) and in [\textit{M. Del Pino}, Sobre un problema cuasilineal de segundo orden. Santiago: Universidad de Chile (Thesis) (1987)] for \(d=1\): any minimizer \(\mathbf{u} \in W_0^{1,p}(\Omega;\mathbb{R}^N)\) of the functional \[ \frac{\int_\Omega (|\nabla v_1|^2 + \cdots + |\nabla v_N|^2)^\frac{p}{2} \,dx}{\int_\Omega (|v_1|^2 + \cdots + |v_N|^2)^\frac{p}{2} \,dx}, \quad v \in W_0^{1,p}(\Omega;\mathbb{R}^N), \quad 1<p<\infty, \] has the form \(\mathbf{u} = \mathbf{c} \omega\), where \(\mathbf{c}\) is a nonzero vector and \(\omega\) is a first eigenfunction of the scalar \(p\)-Laplacian (i.e., \(\omega\) is a minimizer of the same functional in the case \(N=1\)). Here, \(\Omega \subset \mathbb{R}^d\) is a bounded domain. A similar result is proved for the corresponding vectorial fractional Rayleigh quotient.