Higher Robin eigenvalues for the p-Laplacian operator as p approaches 1

Abstract

This work addresses several aspects of the dependence on p of the higher eigenvalues \(\lambda _n\) to the Robin problem,

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta _p u = \lambda |u|^{p-2}u &{} \qquad x\in \Omega ,\\ \ |\nabla u|^{p-2}\dfrac{\partial u}{\partial \nu }+ b |u|^{p-2}u= 0&{}\qquad x\in \partial \Omega . \end{array}\right. } \end{aligned}$$

Here, \(\Omega \subset {{\mathbb {R}}}^N\) is a \(C^1\) bounded domain, \(\nu \) is the outer unit normal, \(\Delta _p u = \text {div}\ (|\nabla u|^{p-2}\nabla u)\) stands for the p-Laplacian operator and \(b\in L^\infty (\partial \Omega )\). Main results concern: (a) the existence of the limits of \(\lambda _n\) as \(p\rightarrow 1\), (b) the ‘limit problems’ satisfied by the ‘limit eigenpairs’, (c) the continuous dependence of \(\lambda _n\) on p when \(1< p <\infty \) and (d) the limit profile of the eigenfunctions as \(p\rightarrow 1\). The latter study is performed in the one dimensional and radially symmetric cases. Corresponding properties on the Dirichlet and Neumann eigenvalues are also studied in these two special scenarios.