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

DOI: http://dx.doi.org/10.12775/TMNA.2004.025

#### 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.

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.

#### Keywords

Nodal solutions; Symmetry; Elliptic semilinear equations

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