On the dependence on \(p\) of the variational eigenvalues of the \(p\)-Laplace operator
It is known that the quasilinear eigenvalue problem \(-\Delta_p u-\mu|u|^{p-2}u\) in \(\Omega\), \(u= 0\) on \(\partial\Omega\), where \(\Omega\) is a bounded domain in \(\mathbb{R}^N\), \(p>1\), and \(\Delta\) is the quasilinear differential operator called \(p\)-Laplacian, has a first positive eigenvalue \(\mu^p>0\), which is simple, having a positive eigenfunction \(u\in W^{1,p}(\Omega)\) (normalized by \(\| u\|_p=1\)). It is also known that \(\mu_1\) as a function of \(p\) is right-continuous, while some condition on \(\partial\Omega\) is required for left-continuity. In this case, we have \[ \lim_{s\to p^-}\mu^s_1= \mu^p_1\tag{1} \] and \[ \lim_{s\to p^-}\int\nabla u_s-\nabla u_p|^p\,dx= 0.\tag{2} \] It was proved by Lindqvist that (2) implies (1), the converse being an open problem. Here, the authors introduce a bigger space and an associated eigenvalue problem as a tool in order to prove (Theorem 4.1) that five other statements are equivalent to (1). In particular, this allows to settle the open problem in the affirmative.
scientific article; zbMATH DE number 6535136
scientific article