Abstract. The equation −Δ p u = u q−1 with zero Dirichlet condition on the boundary is considered in a three-dimensional spherical layer. The existence of arbitrarily many distinct positive solutions in a sufficiently thin layer is proved.

We study the dependence of least nontrivial critical levels of the energy functional corresponding to the zero Dirichlet problem −∆pu = f (u) in a bounded domain Ω ⊂ R N upon domain perturbations. The nonlinearity f is assumed to be superlinear and subcritical. We show that among all (generally eccentric) spherical annuli Ω least nontrivial critical levels attain maximums 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.

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In this note we prove the Payne-type conjecture about the behaviour of the nodal set of least energy sign-changing solutions for the equation −∆pu = f (u) in bounded Steiner symmetric domains Ω ⊂ R N under the zero Dirichlet boundary conditions. The nonlinearity f is assumed to be either superlinear or resonant. In the latter case, least energy sign-changing solutions are second eigenfunctions of the zero Dirichlet p-Laplacian in Ω. We show that the nodal set of any least energy sign-changing solution intersects the boundary of Ω. The proof is based on a moving polarization argument.

On the other hand, r 1 |x|. Thus 2π(R − 1) w 1 4π(R + 1).