Symmetry and monotonicity results for solutions of semilinear PDEs in sector-like domains

Abstract

In this paper we consider semilinear PDEs, with a convex nonlinearity, in a sector-like domain. Using cylindrical coordinates $$(r, \theta , z)$$ ( r , θ , z ) , we investigate the shape of possibly sign-changing solutions whose derivative in $$\theta$$ θ vanishes at the boundary. We prove that any solution with Morse index less than two must be either independent of $$\theta$$ θ or strictly monotone with respect to $$\theta$$ θ . In the special case of a planar domain, the result holds in a circular sector as well as in an annular one, and it can also be extended to a rectangular domain. The corresponding problem in higher dimensions is also considered, as well as an extension to unbounded domains. The proof is based on a rotating-plane argument: a convenient manifold is introduced in order to avoid overlapping the domain with its reflected image in the case where its opening is larger than $$\pi$$ π .