\documentclass[reqno]{amsart} \usepackage{hyperref} \usepackage{amssymb} \AtBeginDocument{{\noindent\small 2007 Conference on Variational and Topological Methods: Theory, Applications, Numerical Simulations, and Open Problems. {\em Electronic Journal of Differential Equations}, Conference 18 (2010), pp. 15--22.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2010 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \setcounter{page}{15} \title[\hfilneg EJDE-2010/Conf/18/\hfil Nodal solutions] {On the number of nodal solutions for a nonlinear elliptic problem on symmetric Riemannian manifolds} \author[M. Ghimenti, A. M. Micheletti\hfil EJDE/Conf/18 \hfilneg] {Marco Ghimenti, Anna Maria Micheletti} % in alphabetical order \address{Marco Ghimenti \newline Dipartimento di Matematica e Applicazioni, Universit\`a di Milano Bicocca, via Cozzi 53, 20125, Milano, Italy} \email{marco.ghimenti@unimib.it} \address{Anna Maria Micheletti \newline Dipartimento di Matematica Applicata, Universit\`a di Pisa, via Buonarroti 1c, 56100, Pisa, Italy} \email{a.micheletti@dma.it} \thanks{Published July 10, 2010.} \subjclass[2000]{35J60, 58G03} \keywords{Riemannian manifolds; nodal solutions; topological methods} \begin{abstract} We consider the problem $$ -\varepsilon^2\Delta_g u+u=|u|^{p-2}u $$ in a symmetric Riemannian manifold $(M,g)$. We give a multiplicity result for antisymmetric changing sign solutions. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{remark}[theorem]{Remark} \newtheorem{definition}[theorem]{Definition} \newtheorem{lemma}[theorem]{Lemma} \section{Introduction} Let $(M,g)$ be a smooth connected compact Riemannian manifold of finite dimension $n\geq 2$ embedded in $\mathbb{R}^N$. We consider the problem \begin{equation}\label{P} %\tag{${\mathscr P}$} -\varepsilon^2\Delta_g u+u=|u|^{p-2}u\text{ in } M, \quad u\in H^1_g(M) \end{equation} where $2
0$ we consider
\begin{gather*}
M_d = \{x\in \mathbb{R}^N:\operatorname{dist}(x,M)\leq d\};\\
M_d^- = \{x\in M:\operatorname{dist}(x,M_\tau)\geq d\}.
\end{gather*}
We choose $d$ small enough such that
\begin{gather*}
G_\tau-\operatorname{cat}_{M_d} M_d=G_\tau-\operatorname{cat}_M M\\
G_\tau-\operatorname{cat}_M M_d^-=G_\tau-\operatorname{cat}_M (M-M_\tau)
\end{gather*}
Now we build two continuous operator
\begin{gather*}
\Phi_\varepsilon^\tau:M_d^-\to {\mathcal{N}}_\varepsilon^\tau
\cap J_\varepsilon^{2(m_\infty+\delta)};\\
\beta:{\mathcal{N}}_\varepsilon^\tau\cap
J_\varepsilon^{2(m_\infty+\delta)}\to M_d,
\end{gather*}
such that $\Phi_\varepsilon^\tau(\tau q)=-\Phi_\varepsilon^\tau(q)$,
$\tau\beta(u)=\beta(-u)$ and
$\beta\circ\Phi_\varepsilon^\tau$ is $G_\tau$ homotopic to the
inclusion $M_d^-\to M_d$.
By equivariant category theory we obtain
\begin{equation}
\begin{aligned}
G_\tau-\operatorname{cat}_M (M-M_\tau)
&=G_\tau-\operatorname{cat}(M_d^-\hookrightarrow M_d)\\
&=G_\tau-\operatorname{cat} \beta\circ\Phi_\varepsilon^\tau\\
&\leq {\mathbb Z}_2-\operatorname{cat} {\mathcal{N}}_\varepsilon^\tau
\cap J_\varepsilon^{2(m_\infty+\delta)}
\end{aligned}
\end{equation}
\section{Technical lemmas}
First of all, we recall that there exists a unique positive
spherically symmetric function $U\in H^1(\mathbb{R}^n)$
such that
\begin{equation}
-\Delta U+U=U^{p-1} \text{ in }\mathbb{R}^n
\end{equation}
It is well known that $U_\varepsilon(x)
=U\left(\frac x\varepsilon\right)$ is a solution of
\begin{equation}
-\varepsilon^2\Delta U_\varepsilon+U_\varepsilon=U_\varepsilon^{p-1}\text{ in }\mathbb{R}^n.
\end{equation}
Secondly, let us introduce the exponential map $\exp:TM\to M$ defined on the tangent
bundle $TM$ of $M$ which is a $C^\infty$ map. Then, for $\rho$ sufficiently small
(smaller than the injectivity radius of $M$ and smaller than $d/2$),
the Riemannian manifold $M$ has a special set of
charts $\{\exp_x:B(0,\rho)\to M\}$.
Throughout the paper we will use the following notation: $B_g(x,\rho)$ is the open ball in $M$
centered in $x$ with radius $\rho$ with respect to the distance given by the metric $g$.
Corresponding to this chart, by choosing an orthogonal coordinate system
$(x_1,\dots,x_n)\subset \mathbb{R}^n$ and identifying $T_xM$ with $\mathbb{R}^n$ for $x\in M$, we can define
a system of coordinates called {\em normal coordinates}.
Let $\chi_\rho$ be a smooth cut off function such that
\begin{gather*}
\chi_\rho(z)=1\quad \text{if }z\in B(0,\rho/2);\\
\chi_\rho(z)=0\quad \text{if }z\in \mathbb{R}^n \setminus B(0,\rho);\\
|\nabla \chi_\rho(z)|\leq2\quad \text{for all } x.
\end{gather*}
Fixed a point $q\in M$ and $\varepsilon>0$,
let us define the function $w_{\varepsilon,q}(x)$ on $M$ as
\begin{equation}
w_{\varepsilon,q}(x)=
\begin{cases}
U_\varepsilon(\exp_q^{-1}(x))\chi_\rho(\exp_q^{-1}(x))
&\text{if }x\in B_g(q,\rho)\\
0&\text{otherwise}
\end{cases}
\end{equation}
For each $\varepsilon>0$ we can define a positive number
$t(w_{\varepsilon,q})$ such that
\begin{equation}
\Phi_\varepsilon(q)=t(w_{\varepsilon,q})w_{\varepsilon,q}
\in H^1_g(M)\cap {\mathcal{N}_\varepsilon}\text{ for }q\in M.
\end{equation}
Namely, $t(w_{\varepsilon,q})$ turns out to verify
\begin{equation}
t(w_{\varepsilon,q})^{p-2}=\frac{\int_M \varepsilon^2|
\nabla_g w_{\varepsilon,q}|^2+|w_{\varepsilon,q}|^2d\mu_g}
{\int_M |w_{\varepsilon,q}|^pd\mu_g}
\end{equation}
\begin{lemma} \label{lem1}
Given $\varepsilon>0$ the application
$\Phi_\varepsilon(q):M\to H^1_g(M)\cap {\mathcal{N}}_\varepsilon$
is continuous. Moreover, given $\delta>0$ there exists
$\varepsilon_0=\varepsilon_0(\delta)$
such that, if $\varepsilon<\varepsilon_0(\delta)$
then $\Phi_\varepsilon(q)\in
{\mathcal{N}}_\varepsilon\cap J_\varepsilon^{m_\infty+\delta}$.
\end{lemma}
For the proof see \cite[ Proposition 4.2]{BBM07}.
Now, fixed a point $q\in M_d^-$, let us define the function
\begin{equation}
\Phi_\varepsilon^\tau(q)=t(w_{\varepsilon,q})w_{\varepsilon,q}-t(w_{\varepsilon,\tau q})w_{\varepsilon,\tau q}
\end{equation}
\begin{lemma}\label{lemma5}
Given $\varepsilon>0$ the application
$\Phi^\tau_\varepsilon(q):M_d^-\to H^1_g(M)\cap {\mathcal{N}}^\tau_\varepsilon$
is continuous. Moreover, given $\delta>0$ there exists $\varepsilon_0=\varepsilon_0(\delta)$
such that, if $\varepsilon<\varepsilon_0(\delta)$
then $\Phi^\tau_\varepsilon(q)\in {\mathcal{N}}^\tau_\varepsilon\cap J_\varepsilon^{2(m_\infty+\delta)}$.
\end{lemma}
\begin{proof}
Since $U_\varepsilon(z)\chi_\rho(z)$ is radially symmetric we set
$U_\varepsilon(z)\chi_\rho(z)=\tilde U_\varepsilon(|z|)$.
We recall that
\begin{gather*}
|\exp^{-1}_{\tau q}\tau x|=d_g(\tau x,\tau q)
=d_g(x,q)=|\exp^{-1}_{q} x|;\\
|\exp^{-1}_{q}\tau x|=d_g(\tau x, q)=d_g(x,\tau q).
\end{gather*}
We have
\begin{align*}
\tau^* \Phi_\varepsilon^\tau(q)(x)
&= -t(w_{\varepsilon,q})w_{\varepsilon,q}(\tau x)
+t(w_{\varepsilon,\tau q})w_{\varepsilon,\tau q}(\tau x) \\
&=-t(w_{\varepsilon,q})\tilde U_\varepsilon(|\exp^{-1}_q(\tau x)|)
+t(w_{\varepsilon,\tau q})
\tilde U_\varepsilon(|\exp^{-1}_{\tau q}(\tau x)|) \\
&=t(w_{\varepsilon,\tau q})\tilde U_\varepsilon(|\exp^{-1}_q(x)|)
-t(w_{\varepsilon,q})
\tilde U_\varepsilon(|\exp^{-1}_{q}(\tau x)|) \\
&=t(w_{\varepsilon,q})\tilde U_\varepsilon(|\exp^{-1}_q(x)|)-
t(w_{\varepsilon,q})\tilde U_\varepsilon(|\exp^{-1}_{\tau q}(x)|),
\end{align*}
because by the definition we have $t(w_{\varepsilon,q})
=t(w_{\varepsilon,\tau q})$.
Moreover by definition the support of the function
$\Phi_\varepsilon^\tau$ is
$B_g(q,\rho)\cup B_g(\tau q,\rho)$, and
$B_g(q,\rho)\cap B_g(\tau q,\rho)=\emptyset$
because $\rho