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

Abstract

Let $B_1$ be a ball in ${\mathbb{R}}^N$ centred at the origin and $B_0$ be a smaller ball compactly contained in $B_1$. For $p\in (1,\infty )$, using the shape derivative method, we show that the first eigenvalue of the $p$-Laplacian in annulus $B_1\setminus \overline{B_0}$ strictly decreases as the inner ball moves towards the boundary of the outer ball. The analogous results for the limit cases as $p\to 1$ and $p\to \infty$ 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 $p$-Laplacian on bounded radial domains.

19 pages