On qualitative properties of solutions for elliptic problems with the $p$-Laplacian through domain perturbations

Abstract

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.