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.