  We study the dependence of least nontrivial critical levels of the energy
functional corresponding to the zero Dirichlet problem $-\Delta_p u = f(u)$ in
a bounded domain $\Omega \subset \mathbb{R}^N$ upon domain perturbations.
Assuming that the nonlinearity $f$ is superlinear and subcritical, we establish
Hadamard-type formulas for such critical levels. As an application, we show
that among all (generally eccentric) spherical annuli $\Omega$ least nontrivial
critical levels attain maximum if and only if $\Omega$ is concentric. As a
consequence of this fact, we prove the nonradiality of least energy nodal
solutions whenever $\Omega$ is a ball or concentric annulus.
