  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 Fu\v{c}ik spectrum of the
$p$-Laplacian on bounded radial domains.
