Existence, unique continuation and symmetry of least energy nodal solutions to sublinear Neumann problems

We consider the sublinear problem {-Delta u = vertical bar u vertical bar(q-2)u in Omega, u(v) = 0 on partial derivative Omega, where Omega subset of R-N is a bounded domain, and 1 <= q < 2. For q - 1, vertical bar u vertical bar(q-2)u will be identified with sgn (u). We establish a variational principle for least energy nodal solutions, and we investigate their qualitative properties. In particular, we show that they satisfy a unique continuation property (their zero set is Lebesgue-negligible). Moreover, if Omega is radial, then least energy nodal solutions are foliated Schwarz symmetric, and they are nonradial in case Omega is a ball. The case q=1 requires special attention since the formally associated energy functional is not differentiable, and many arguments have to be adjusted.