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

Abstract

We consider the sublinear problem \begin {equation*} \left\{\begin{array}{r c l c} -\Delta u & = &|u|^{q-2}u & \textrm{in }\Omega, \\ u_n & = & 0 & \textrm{on }\partial\Omega,\end{array}\right. \end {equation*} where $\Omega \subset \real^N$ is a bounded domain, and $1 \leq q < 2$. For $q=1$, $|u|^{q-2}u$ 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 treatment since the formally associated energy functional is not differentiable, and many arguments have to be adjusted.