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

Original · EN

We consider the sublinear problem equation* {arrayr c l c -Δu & = &|u|q⁻²u & in Ω, uₙ & = & 0 & on ∂Ω,array. equation* where Ω⊂ ⁿ is a bounded domain, and 1 ≤ q < 2. For q=1, |u|q⁻²u will be identified with (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 Ω is radial, then least energy nodal solutions are foliated Schwarz symmetric, and they are nonradial in case Ω is a ball. The case q=1 requires special treatment since the formally associated energy functional is not differentiable, and many arguments have to be adjusted.