\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2009(2009), No. 40, pp. 1--5.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2009 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2009/40\hfil Non-radially symmetric solutions]
{A note on nodal non-radially symmetric solutions to
Emden-Fowler equations}

\author[M. Ramos, W. Zou\hfil EJDE-2009/40\hfilneg]
{Miguel Ramos, Wenming Zou}  % in alphabetical order

\address{Miguel Ramos  \newline 
Universidade de Lisboa, CMAF-Faculty  of Science\\
 Av. Prof. Gama Pinto, 2, 1649-003-Lisboa, Portugal}
\email{mramos@ptmat.fc.ul.pt}

\address{Wenming Zou \newline
 Department of Mathematical Sciences, Tsinghua University\\
Beijing 100084, China}
\email{wzou@math.tsinghua.edu.cn}

\thanks{Submitted February 19, 2009. Published March 19, 2009.}

\subjclass[2000]{35J20, 35J25, 35B99}
\keywords{Emden-Fowler equation; nodal solutions; symmetric solutions;
 \hfill\break\indent variational methods}

\begin{abstract}
 We prove the existence  of an unbounded  sequence of
 sign-changing and non-radially  symmetric solutions to the problem
 $-\Delta u = |u|^{p-1}u$ in $\Omega$, $u =  0$ on  $\partial\Omega$,
 $u(gx)= u(x$),   $ x\in \Omega$, $g\in G$, where $\Omega$ is
 an annulus of $\mathbb{R}^N$ ($N\geq 3$), $1<p< (N+2)/(N-2)$ and
 $G$  is a non-transitive closed subgroup of the orthogonal group $O(N)$.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\numberwithin{equation}{section}

\section{Introduction}

In this note we consider  the  sign-changing and non-radially
symmetric solutions to the following  Emden-Fowler equation:
\begin{gather}
  -\Delta u = |u|^{p-1}u,   \quad  x\in \Omega,\label{group1}\\
   u =  0,   \quad  x\in \partial\Omega,  \label{group2} \\
  u(gx)  = u(x),  \quad   x\in \Omega, \;g\in G, \label{group3}
\end{gather}
where $\Omega$ is a   unit  ball $\Omega:=\{x\in \mathbb{R}^N: |x|<1\}$ or
an annulus  $\Omega:=\{x\in \mathbb{R}^N: a<|x|<b\}$, $ 0<a<b$, $N\geq 3$,
$1<p<(N+2)/(N-2)$ and $G$  is a closed subgroup  of   the orthogonal
group $O(N)$ of degree  $N$.   Here $gx$  is the product of the
column  vector $x$  and the   matrix $g$ and  a solution  of
\eqref{group1}-\eqref{group3} will be called a $G$-invariant solution.

It is known  that \eqref{group1}-\eqref{group2}   has infinitely many
sign-changing radially   symmetric solutions  when $1<p<(N+2)/(N-2)$
(cf.   \cite{kajikia-2, kajikia-3, niwm, struwe}) and each one of
them has   finitely many zero points. The existence of sign-changing
solutions of  \eqref{group1}-\eqref{group2}  with further information  on
the nodal domains is considered  in \cite{Bartsch-JFA}  but no conclusions
on the non-radial symmetry are derived.

Clearly, a radially symmetric
solution is a $G$-invariant solution, for any
subgroup $G$ of $O(N)$. The converse problem was considered in
\cite{kajikia} where the author proved that there exist  solutions
which are  $G$-invariant and not radially symmetric if  $G$ is  not
transitive on $S^{N-1}:=\{x\in \mathbb{R}^N: |x|=1\}$. In the sequel, we
say that  $G$ is transitive if for any two points $x, y\in S^{N-1}$
there exists a $g\in G$ such that $y=gx$.  Under this assumption, in   \cite[Theorem 1]{kajikia} it is proved that the problem \eqref{group1}-\eqref{group3} admits an unbounded sequence of  $G$-invariant and non-radially symmetric solutions. According to a celebrated theorem by Gidas, Ni and Nirenberg \cite{gidas-ni-nirenberg}, non-radially symmetric solutions must change their sign if the domain is a ball.  In this note we derive the conclusion that these solutions must indeed change sign in $\Omega$, even if the domain is an annulus. Precisely, we prove the following.

\begin{theorem} \label{group-th1}
If  $G$  is not  transitive  on
$S^{N-1}$,   then there exists a sequence $\{w_k\}$ of  solutions of
 \eqref{group1}-\eqref{group3}  such that  each  $w_k$  is
$G$-invariant, sign-changing  and  non-radially  symmetric.
Moreover,    $\|w_k\|\to \infty$ as $k\to \infty $.
\end{theorem}

We denote by $\|\cdot\|$ the usual norm in $H_0^1(\Omega)$. We
mention that, by construction, the solutions $w_k$ have a
well-determined Morse index (cf. \cite{ghoussoub, ramos-zou}), so
that it is likely that further conclusions on their nodal domains
can be derived, in the line of the work in \cite{yang}.

We recall from \cite[Corollaries 1 and 2]{kajikia} that Theorem
\ref{group-th1} applies in case $G$ is finite or has dimension not
greater than $N-2$. A typical example is  $G=\{Id, -Id\}$, where
$Id$ is the  unit matrix. It follows that
\eqref{group1}-\eqref{group3} has infinitely many {\em
sign-changing} non-radially  symmetric  and even solutions.
Another example is
$$
G=\big\{\begin{pmatrix}
e & 0 \\
0 & w
\end{pmatrix} :  e\in O(m),\;\; w\in O(N-m)\big\}, \quad
1\leq m<N.
$$
Then by Theorem  \ref{group-th1}, \eqref{group1}-\eqref{group3}
has a sequence of solutions $\{u_k\}$ such that each $u_k$  is
sign-changing  and $u_k(x)=u_k(|x'|, |x''|)$ for all
$x'\in\mathbb{R}^m, x''\in \mathbb{R}^{N-m}$ with $x=(x', x'')\in
\Omega$, but $u_k(x)\not=u_k(|x|)$.

The proof of Theorem \ref{group-th1} is given in the next section.
We combine the approach in \cite{kajikia} (namely the crucial
estimates in Lemmas \ref{lemma-group-1} and \ref{lemma-groupee})
with the method introduced in \cite{ramos-zou} for finding
sign-changing solutions to superlinear elliptic equations such as
the one in (\ref{group1}), which is essentially contained in the
strict inequality (\ref{SSKK}) below.

\section{Proof of Theorem \ref{group-th1}}

Let
$$
H_0^1(\Omega, G)=\{u\in H_0^1(\Omega): u(gx)=u(x), x\in \Omega,
g\in G\}
$$
equipped with  the  inner  product
$\langle u, v \rangle=\int_\Omega \nabla u\cdot \nabla v dx$ and the
corresponding norm   $\|u\|= \langle u, u \rangle^{1/2}$. We also
denote by $\|u\|_{p+1}$ the $L^{p+1}(\Omega)$ norm of $u$.
Solutions of \eqref{group1}-\eqref{group3} are   critical points
of the functional $I$ defined by
$$
I(u)=\frac{1}{2}\|u\|^2-\frac{1}{p+1}\|u\|_{p+1}^{p+1},\quad
u\in  H_0^1(\Omega, G).
$$
We  denote  by $\{\lambda_k(\Omega, G)\}_ {k\in \mathbb{N}}$  the
increasing sequence of eigenvalues of the problem
\begin{equation}\label{eigen-group-111}
\begin{gathered}
-\Delta u  = \lambda u,  \quad x\in \Omega, \\
 u  =  0,  \quad x\in \partial\Omega,   \\
 u(gx)   =  u(x),  \quad  x\in \Omega, \; g\in  G.
\end{gathered}
\end{equation}


\begin{lemma}[cf.  \cite{kajikia, kajikia-2,kajikia-3,niwm}]
 \label{lemma-group-1}
 The set of  radially  symmetric critical  points of $I$
consists of a sequence $\{\pm u_k\}_ {k\in \mathbb{N}}$ and the zero  solution.
Moreover,
$$
0<\beta_1<\beta_2<\dots<  \beta_k<\dots\to \infty, \quad
\text{where } \beta_k=I(\pm u_k),
$$
and  there  exists $A_0>0$ independent of $k$   such that
$$
A_0  k^{\frac{2(p+1)}{p-1}}\leq \beta_k, \quad  k\in \mathbb{N}.
$$
\end{lemma}

Let
$$
G(x)=\{gx:  g\in G\}, \quad   x\in S^{N-1}.
$$
Then $G(x)$  is a closed  submanifold of  $S^{N-1}$ and we denote  by
$\dim G(x)$ its  dimension,  so that
$0\leq  \dim G(x)\leq N-1$. Let
 $$
 m:=m(G):=\max\{\dim G(x):  x\in  S^{N-1}\}.
$$

\begin{lemma}[cf.  \cite{kajikia}] \label{lemma-groupee}
Assume that $G$  is not transitive on $S^{N-1}$. Then $0\leq m\leq N-2$
and  there exists a positive constant $C_1$ independent of $k$ such that
$$
\lambda_k(\Omega, G)\leq C_1  k^{\frac{2}{N-m}}, \quad k\in\mathbb{N}.
$$
\end{lemma}

Now, let $E_k$ be the eigenspace associated to the eigenvalues
$\lambda_i(\Omega, G)$ with $i=1,\ldots,k$
and $S_k:=\{u\in E_{k-1}^\perp:\|u\|_{p+1}=1\}$. As observed in
 \cite{kajikia}, it follows from Lemma  \ref{lemma-groupee} that
\begin{equation} \label{sup_{E_k}Ileq}
\sup_{E_k}I   \leq B_0 k^{\frac{2(p+1)}{(N-m)(p-1)}},
\end{equation}
while a simple computation  shows that
\begin{equation}\label{inf_{S_k}Igeq}
\inf_{S_k}  I\geq
B_1 \lambda_k(\Omega, G)^{\alpha}-B_2,
\end{equation}
for some positive constants $ B_0$, $  B_1$, $B_2$
independent of $k$, where $\alpha$ is given by
$\alpha=\frac{(2+N)-p(N-2)}{2(p+1)}>0$.

By observing that  $I(u)>0$ if $u$ is a
nontrivial critical point of $I$, we define
\begin{align*}%\label{box}
 N_1:=\sup \big\{& c\in \mathbb{R} : c>0
\text{ is a critical value  of $I$  corresponding to  $G$-invariant }  \\
&\text{sign-changing and non-radially symmetric critical points} \big\}
\end{align*}
(We set $N_1=0$ in case this set is empty).
To prove Theorem  \ref{group-th1} we must show   that  the above set
is nonempty  and that $N_1=\infty$. In the sequel we argue by
contradiction by assuming that  $N_1<\infty$.

According to (\ref{inf_{S_k}Igeq}), we can fix  $k_0>0$  such that
\begin{equation}\label{kkkkk}
   \inf_{S_k}  I >N_1 \quad \text{for all } k\geq k_0.
\end{equation}
Let
$$
 N_2:=\max \{k\in \mathbb{N}: A_0  (k-k_0+1)^{\frac{2(p+1)}{p-1}}\leq  B_0
k^{\frac{2(p+1)}{(N-m)(p-1)}}\}.
$$
Thanks to Lemma \ref{lemma-groupee}, $N_2$ is finite.
We choose
$k^\ast$  large  enough  such that $k^\ast>\max\{k_0,N_2\}$.
From now on we only consider the integers $k$ lying in the interval
$[k_0,k^\ast]$.
Let
\begin{equation}\label{ccccc}
 C^\ast=\sup_{E_{k^\ast}} I < \infty.
\end{equation}
We also fix $R_k>0$ in such a way that
$$
\|u\|_{p+1}>1 , \quad  I(u)<0 \quad \text{for all } u\in E_k
\text{ with } \|u\|\geq R_k.
$$
We may assume that $R_k$ increases with $k$.

 Let $P$  denote the positive cone of $H_0^1(\Omega, G)$, that
is $P:= \{ u\in  H_0^1(\Omega, G):  u(x)\geq 0, x\in \Omega\}$. It
follows from \cite[Lemma 2.4]{ramos-zou} that
\begin{equation}\label{SSKK}
\mathop{\rm dist}\Big(\big(\cup_{k=
k_0}^{k^\ast} S_k\big)\cap I^{C^\ast}, \pm P\Big) >0,
\end{equation}
where $ I^{C^\ast}:=\{u: I(u)\leq C^\ast\}$. Let
$D:=\{u\in H_0^1(\Omega, G):  \mathop{\rm dist}(u, P)<\varepsilon_0\}$,
$D^\ast:=-D\cup D$,
${\mathcal{U}}:=E\backslash D^\ast$.   Then,
for  $\varepsilon_0$ small enough, we  have
 \begin{equation} \label{bigcup}
\big(\cup_{k= k_0}^{k^\ast} S_k\big)\cap I^{C^\ast}\subset
{\mathcal{U}}.
\end{equation}
   Moreover, as shown in \cite{conti},
$D^\ast\cap{\mathcal{K}}\subset (-P\cup P)$, where
${\mathcal{K}}:=\{u\in H_0^1(\Omega, G):  I'(u)=0\}$.
 For $k\in [k_0, k^\ast], $ we  set
\begin{gather*}
T_k:=\{h:  h\in C(\Theta_k, E), h \text{ is odd }, h(u)=u \text{ on
}  \partial \Theta_k\}, \\
 \Theta_k:=\{u\in E_k:  \|u\|< R_k\}, \quad
\partial \Theta_k:=\{u\in E_k:  \|u\|= R_k\}.
\end{gather*}
Define
\begin{equation}\label{zzzz}
\begin{aligned}
Z_k:=\big\{&h(\overline{\Theta_i\backslash A}):   h\in T_i,\;
i\in [k, k^\ast],\; A\in  {\mathcal{E}}, \\
& \gamma (A)\leq i-k, \; I(h(\overline{\Theta_i\backslash
A}))\leq C^\ast \big\},
\end{aligned}
\end{equation}
 where ${\mathcal{E}}$ is the
family of closed subsets $A$  of $H_0^1(\Omega, G)$  such that
$0\not\in A$ and  $-u\in A$ whenever $u\in A$; $\gamma(A)$ denotes
the genus of $A$. Clearly, $Z_k\not=\emptyset$  since $Id\in T_k$; also,
$Z_{k+1}\subset Z_k$.

\begin{lemma} \label{stst-1}
 $  B\cap \mathcal{U}\cap S_k\not=\emptyset$ for
any $B\in Z_k$.
\end{lemma}


\begin{proof}
  Thanks to \eqref{bigcup} it is sufficient to prove that
$ B\cap S_k \not=\emptyset$.
  This, in turn, can be derived in a standard way. For completeness,
we sketch the argument as in \cite[Proposition 9.23]{Rab86}.
We write $B=h(\overline{\Theta_i\backslash A})$  with
$h\in T_i, k^\ast\geq i\geq k$ and $\gamma(A)\leq i-k$. Let
$W_1:=\{u\in \Theta_i: \|h(u)\|_{p+1}<1\}$ and
$W_2:=\{u\in \Theta_i: \|h(u)\|_{p+1}=1\}$. Then $W_1$  is a symmetric
bounded neighborhood of 0 in $\Theta_i$ and hence $\gamma(\partial W_1)=i$,
while $\partial W_1\subset W_2$ by our choice of $R_k$.
Thus $\gamma(W_2)\geq i$ and so  $\gamma(h(\overline{W_2\backslash
A}))\geq \gamma(\overline{W_2\backslash A})\geq k>k-1$.
Hence $ h(\overline{W_2\backslash A})\cap
E_{k-1}^\perp\not=\emptyset$  and this proves the claim.
\end{proof}

 Now, for   $k_0\leq k\leq k^\ast$ we define
$$
c_k=\inf_{B\in Z_k}\max_{u\in B\cap {\mathcal{U}}}I(u).
$$
 Thanks to (\ref{kkkkk}) and  Lemma
\ref{stst-1}, $ c_k$ is well defined and  $c_k\geq
\inf_{S_k}I>N_1$.   Clearly,
$c_{k_0}\leq c_{k_0+1}\leq \dots\leq c_{k^\ast}$.


\begin{lemma} \label{stst-www}
If  $c_k=c_{k+1}=\dots=c_{k+\ell}=:c, $ then
$\gamma({\mathcal{K}}_c\cap {\mathcal{U}})\geq \ell+1$, where
${\mathcal{K}}_c:=\{u\in H_0^1(\Omega, G):   I(u)=c, I'(u)=0\}$.
\end{lemma}

\begin{proof}
In view of a contradiction, assume that
$ \gamma({\mathcal{K}}_c\cap {\mathcal{U}})\leq\ell $. Since
$ {\mathcal{K}}_c^s:={\mathcal{K}}_c\cap {\mathcal{U}}$ is compact and
$0\not\in  {\mathcal{K}}_c^s$, there exists a closed  neighborhood
$U$ of  ${\mathcal{K}}_c^s$  such that $\gamma(U)\leq  \ell$.
Let $V$  be an open   neighborhood
 of ${\mathcal{K}}_c\cap {(-P\cup P)}:={\mathcal{K}}_c^{pn}$ such that
$V\subset {{D}}^\ast$.
The well-known deformation   lemma  implies that for $\varepsilon>0$ small enough we
 can   find  a flow  $\eta\in C([0, 1]\times E, E)$  such that
   $\eta(1, u)$ is odd in $u$, $\eta(1, I^{c+\varepsilon}\backslash
 (\buildrel\circ\over U\cup V))\subset  I^{c-\varepsilon} $
and $\eta(1, \cdot)=Id$ on $\partial\Theta_i$  for
 $i\in [k, k^\ast]$ (here we use the fact that  $I<0$ on  $\partial\Theta_i$ and $c>N_1\geq 0$). Moreover,  the  flow
 $\eta$  keeps  $\pm {{D}}$ invariant, that is $\eta(t, \pm
 {{D}})\subset  \pm {{D}}$ for every $t$ (see for example  \cite{blw0, conti, ramos-zou, mwen18}). Hence, $\eta(1, I^{c+\varepsilon}\backslash
 \buildrel\circ\over U)\subset  I^{c-\varepsilon}\cup {{D}}^\ast$. Choose  $B\in
 Z_{k+ \ell}$  such that $\max_{B\cap {\mathcal{U}}}I\leq c+\varepsilon$,
 $B=h(\overline{\Theta_i\backslash A})$  with $h\in T_i, i\in
 [k+ \ell, k^\ast], \gamma(A)\leq i-(k+ \ell), \sup_BI\leq C^\ast$. Similarly to  \cite[Proposition 9.18]{Rab86} we find that $\overline{B\backslash U}\in
  Z_k$.   Since $\eta$ is a descending flow,  also $\eta(1, \overline{B\backslash U})\in Z_k$.  But
$\eta(1, \overline{B\backslash U})\cap {\mathcal{U}}
=\eta(1, \mathcal{U}\cap \overline{B\backslash U})\cap {\mathcal{U}}\subset
\eta(1, I^{c+\varepsilon} \backslash \buildrel\circ\over U)\cap
  {\mathcal{U}} \subset  (I^{c-\varepsilon}\cup{ {D}}^\ast)  \cap
  {\mathcal{U}} \subset   I^{c-\varepsilon}$. This contradicts the definition of $c$ and proves the lemma.
\end{proof}



\begin{proof}[Proof of Theorem  \ref{group-th1}]
  Thanks to Lemma  \ref{stst-www}, we can conclude similarly to \cite{kajikia}, and so we only sketch the argument. Since $c_k>N_1$
for all $k\in [k_0, k^\ast]$, by Lemma \ref{lemma-group-1}, we see
that $\{c_{k_0}, c_{k_0+1}, \dots, c_{k^\ast}\}\subset \{\beta_1,
\beta_2, \dots \}$.   Assume  $c_k=c_{k+1}$ for some $k\in [k_0,
k^\ast-1]$.  Then, by Lemma  \ref{stst-www},
$\gamma({\mathcal{K}}_{c_k}\cap {\mathcal{U}})\geq 2$.  But
$c_k=\beta_i$  for some  $i$, and so  ${\mathcal{K}}_{c_k}\cap
{\mathcal{U}}=\{u_i, -u_i\}$. This is a contradiction and it follows that $\{c_k\}_{k=k_0}^{k^\ast}$ is   strictly increasing.
Therefore,  $c_{k^\ast}=\beta_j$ for some $j\geq k^\ast-k_0+1$.  Hence, by Lemma
\ref{lemma-group-1} and (\ref{sup_{E_k}Ileq}),
$$
A_0  (k^\ast-k_0+1)^{\frac{2(p+1)}{p-1}}
\leq \beta_j= c_{k^\ast}\leq  B_0
(k^\ast)^{\frac{2(p+1)}{(N-m)(p-1)}}.
$$
The very definition of $N_2$ implies $ k^\ast \leq N_2$.
This contradicts our choice of $k^\ast$ and proves our claim that
$N_1=\infty$.
\end{proof}

\subsection*{Acknowledgments}
M. Ramos was supported  by FCT, program POCI-ISFL-1-209
(Portugal/Feder-EU). W. Zou was supported by NSFC (10871109, 10571096),
 SRF-ROCS-SEM and the program  of the  Ministry of Education
 in  China for  NCET in Universities of China.

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