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

This work addresses several aspects of the dependence on p of the higher eigenvalues \(\lambda _n\) ${\lambda }_n$ to the Robin problem, \(\begin{aligned} {\left\{ \begin{array}{ll} -\Delta _p u = \lambda |u|^{p-2}u &{} \qquad x\in \Omega ,\\ \ |\nabla u|^{p-2}\dfrac{\partial u}{\partial \nu }+ b |u|^{p-2}u= 0&{}\qquad x\in \partial \Omega . \end{array}\right. } \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}-{\mathrm{\Delta }}_{p}u=\lambda {|u|}^{p-2}u\hfill & x\in \mathrm{\Omega },\hfill \\ |\mathrm{\nabla }u|_{}^{p-2}{\displaystyle \frac{\partial u}{\partial \nu }}+b{|u|}^{p-2}u=0\hfill & x\in \partial \mathrm{\Omega }.\hfill \end{array}\right)\end{array}$$ Here, \(\Omega \subset {{\mathbb {R}}}^N\) $\mathrm{\Omega }\subset {\mathbb{R}}^N$ is a \(C^1\) $C^1$ bounded domain, \(\nu \) $\nu$ is the outer unit normal, \(\Delta _p u = \text {div}\ (|\nabla u|^{p-2}\nabla u)\) ${\mathrm{\Delta }}_{p}u={\text{div}(|\mathrm{\nabla }u|}^{p-2}\mathrm{\nabla }u)$ stands for the p-Laplacian operator and \(b\in L^\infty (\partial \Omega )\) $b\in L^{\infty }(\partial \mathrm{\Omega })$ . Main results concern: (a) the existence of the limits of \(\lambda _n\) ${\lambda }_n$ as \(p\rightarrow 1\) $p\to 1$ , (b) the ‘limit problems’ satisfied by the ‘limit eigenpairs’, (c) the continuous dependence of \(\lambda _n\) ${\lambda }_n$ on p when \(1< p <\infty \) $1<p<\infty$ and (d) the limit profile of the eigenfunctions as \(p\rightarrow 1\) $p\to 1$ . 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.