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 −Δ_p u=f(u) in a bounded domain Ω⊂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 Ω least nontrivial critical levels attain maximum if and only if Ω is concentric. As a consequence of this fact, we prove the nonradiality of least energy nodal solutions whenever Ω is a ball or concentric annulus.