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

Abstract

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