On the strict monotonicity of the first eigenvalue of the 𝑝-Laplacian on annuli

Abstract

Let B 1 B_1 be a ball in R N \mathbb {R}^N centred at the origin and let B 0 B_0 be a smaller ball compactly contained in B 1 B_1 . For p ∈ ( 1 , ∞ ) p\in (1, \infty ) , using the shape derivative method, we show that the first eigenvalue of the p p -Laplacian in annulus B 1 ∖ B 0 ¯ B_1\setminus \overline {B_0} strictly decreases as the inner ball moves towards the boundary of the outer ball. The analogous results for the limit cases as p → 1 p \to 1 and p → ∞ p \to \infty are also discussed. Using our main result, further we prove the nonradiality of the eigenfunctions associated with the points on the first nontrivial curve of the Fučik spectrum of the p p -Laplacian on bounded radial domains.