A remark on minimal nodal solutions of an elliptic problem in a ball

Abstract

Consider the equation $-\Delta u = u_{+}^{p-1}-u_{-}^{q-1}$ in the unit ball $B$ with a homogeneous Dirichlet boundary condition. We assume $2< p,q< 2^{*}$. Let $\varphi(u)=(1/2)\int_{B} |\nabla u|^{2} dx-(\1/p)\int_{B}u_{+}^{p}dx -(1/q)\int_{B}u_{-}^{q}dx$ be the functional associated to this equation. The nodal Nehari set is defined by $\mathcal M=\{u\in H^{1}_{0}(B): u_{+}\neq 0,\ u_{-}\neq 0,\ \langle\varphi'(u_{+}),u_{+}\rangle= \langle\varphi'(u_{-}),u_{-}\rangle=0\}$. Now let $\mathcal M_{\text{\rm rad}}$ denote the subset of $\mathcal M$ consisting of radial functions and let $\beta_{\text{\rm rad}}$ be the infimum of $\varphi$ restricted to $\mathcal M_{\text{\rm rad}}$. Furthermore fix two disjoint half balls $B^{+}$ and $B^{-}$ and denote by $\mathcal M_{h}$ the subset of $\mathcal M$ consisting of functions which are positive in $B^{+}$ and negative in $B^{-}$. We denote by $\beta_{h}$ the infimum of $\varphi$ restricted to $\mathcal M_{h}$. In this note we are interested in obtaining inequalities between $\beta_{\text{\rm rad}}$ and $\beta_{h}$. This problem is related to the study of symmetry properties of least energy nodal solutions of the equation under consideration. We also consider the case of the homogeneous Neumann boundary condition.