\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 $$\label{P} %\tag{{\mathscr P}} -\varepsilon^2\Delta_g u+u=|u|^{p-2}u\text{ in } M, \quad u\in H^1_g(M)$$ where $20$ 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{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} \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 $$-\Delta U+U=U^{p-1} \text{ in }\mathbb{R}^n$$ It is well known that $U_\varepsilon(x) =U\left(\frac x\varepsilon\right)$ is a solution of $$-\varepsilon^2\Delta U_\varepsilon+U_\varepsilon=U_\varepsilon^{p-1}\text{ in }\mathbb{R}^n.$$ 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 $$w_{\varepsilon,q}(x)= \begin{cases} U_\varepsilon(\exp_q^{-1}(x))\chi_\rho(\exp_q^{-1}(x)) &\text{if }x\in B_g(q,\rho)\\ 0&\text{otherwise} \end{cases}$$ For each $\varepsilon>0$ we can define a positive number $t(w_{\varepsilon,q})$ such that $$\Phi_\varepsilon(q)=t(w_{\varepsilon,q})w_{\varepsilon,q} \in H^1_g(M)\cap {\mathcal{N}_\varepsilon}\text{ for }q\in M.$$ Namely, $t(w_{\varepsilon,q})$ turns out to verify $$t(w_{\varepsilon,q})^{p-2}=\frac{\int_M \varepsilon^2| \nabla_g w_{\varepsilon,q}|^2+|w_{\varepsilon,q}|^2d\mu_g} {\int_M |w_{\varepsilon,q}|^pd\mu_g}$$ \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 $$\Phi_\varepsilon^\tau(q)=t(w_{\varepsilon,q})w_{\varepsilon,q}-t(w_{\varepsilon,\tau q})w_{\varepsilon,\tau q}$$ \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 $\rho0$ there exists $\varepsilon_0(\delta)$ such that, for $\varepsilon<\varepsilon_0(\delta)$ $$2m_\varepsilon\leq m_\varepsilon^\tau \leq 2J_\varepsilon(\Phi_\varepsilon(q))\leq 2(m_\infty+\delta).$$ Since $\lim_{\varepsilon\to0}m_\varepsilon=m_\infty$ (see \cite[Remark 5.9]{BBM07}) we get the claim. \end{proof} For any function $u\in {\mathcal{N}}_\varepsilon^\tau$ we can define a point $\beta(u)\in \mathbb{R}^N$ by $$\beta(u)=\frac{\int_M x|u^+(x)|^pd\mu_g}{ \int_M |u^+(x)|^pd\mu_g}$$ \begin{lemma}\label{lemma7} There exists $\delta_0$ such that, for any $0<\delta<\delta_0$ and any $0<\varepsilon<\varepsilon_0(\delta)$ (as in Lemma \ref{lemma5}) and for any function $u\in {\mathcal{N}}_\varepsilon^\tau\cap J_\varepsilon^{2(m_\infty+\delta)}$, it holds $\beta(u)\in M_d$. \end{lemma} \begin{proof} Since $\tau^* u=u$ we set $M^+=\{x\in M:u(x)>0\},\quad M^-=\{x\in M:u(x)<0\}.$ It is easy to see that $\tau M^+=M^-$. Then we have \begin{align*} J_\varepsilon(u) &=\Big(\frac 12-\frac 1p\Big)\frac 1{\varepsilon^n} \int_M|u|^pd\mu_g \\ &=\Big(\frac 12-\frac 1p\Big)\frac 1{\varepsilon^n} \Big[\int_{M^+}|u^+|^pd\mu_g+\int_{M^-}|u^-|^pd\mu_g \Big]=2J_\varepsilon(u^+) \end{align*} By the assumption $J_\varepsilon(u)\leq 2(m_\infty+\delta)$ we have $J_\varepsilon(u^+)\leq m_\infty+\delta$ then by Proposition 5.10 of \cite{BBM07} we get the claim. \end{proof} \begin{lemma} There exists $\varepsilon_0>0$ such that for any $0<\varepsilon<\varepsilon_0$ the composition $$I_\varepsilon=\beta\circ\Phi_\varepsilon^\tau:M_d^-\to M_d \subset\mathbb{R}^N$$ is well defined, continuous, homotopic to the identity and $I_\varepsilon(\tau q)= \tau I_\varepsilon(q)$. \end{lemma} \begin{proof} It is easy to check that $\Phi_\varepsilon^\tau(\tau q)=-\Phi_\varepsilon^\tau(q),\quad \beta(-u)=\tau\beta(u).$ Moreover, by Lemma \ref{lemma5} and by Lemma \ref{lemma7}, for any $q\in M_d^-$ we have $\beta\circ\Phi_\varepsilon^\tau(q)=\beta(\Phi_\varepsilon(q))\in M_d$, and $I_\varepsilon$ is well defined. In order to show that $I_\varepsilon$ is homotopic to identity, we evaluate the difference between $I_\varepsilon$ and the identity as follows. \begin{align*} I_\varepsilon(q)-q &=\frac{\int_M(x-q)|w^+_{\varepsilon,q}|^pd\mu_g} {\int_M|w^+_{\varepsilon,q}|^pd\mu_g}\\ &=\frac{\int_{B(0,\rho)} z\left|U\left(\frac z\varepsilon\right)\chi_\rho(|z|)\right|^p \big|g_q(z)\big|^{1/2}} { \int_{B(0,\rho)} \left|U\left(\frac z\varepsilon\right)\chi_\rho(|z|)\right|^p \big|g_q(z)\big|^{1/2}} \\ &= \frac{ \varepsilon\int_{B(0,\rho/\varepsilon)}z \big|U(z)\chi_\rho(|\varepsilon z|)\big|^p \big|g_q(\varepsilon z)\big|^{1/2}} {\int_{B(0,\rho/\varepsilon)} \big|U(z)\chi_\rho(|\varepsilon z|)\big|^p \big|g_q(\varepsilon z)\big|^{1/2}}, \end{align*} hence $|I_\varepsilon(q)-q| <\varepsilon c(M)$ for a constant $c(M)$ that does not depend on $q$. \end{proof} Now, by previous lemma and by Theorem \ref{castroclapp} we can prove Theorem \ref{mainteo}. In fact, we know that, if $\varepsilon$ is small enough, there exist $G_\tau-\operatorname{cat}(M-M_\tau)$ minimizers which change sign, because they are antisymmetric. We have only to prove that any minimizer changes sign exactly once. Let us call $\omega=\omega_\varepsilon$ one of these minimizers. Suppose that the set $\{x\in M:\omega_\varepsilon(x)>0 \}$ has $k$ connected components $M_1, \dots, M_k$. Set $$\omega_i=\begin{cases} \omega_\varepsilon(x) & x\in M_i\cup\tau M_i;\\ 0 &\text{elsewhere} \end{cases}$$ For all $i$, $\omega_i\in {\mathcal{N}}^\tau_\varepsilon$. Furthermore we have $$J_\varepsilon(\omega)=\sum_iJ_\varepsilon(\omega_i),$$ thus $$m_\varepsilon^\tau=J_\varepsilon(\omega) =\sum_{i=1}^k J_\varepsilon(\omega_i)\geq k \cdot m_\varepsilon^\tau,$$ so $k=1$, that concludes the proof. \begin{thebibliography}{00} \bibitem{BBM07} V.~Benci, C.~Bonanno, and A.M.~Micheletti; \emph{On the multiplicity of solutions of a nonlinear elliptic problem on {R}iemannian manifolds}, J. Funct. Anal. \textbf{252} (2007), no.~2, 464--489. \bibitem{BC91} V.~Benci and G.~Cerami; \emph{The effect of the domain topology on the number of positive solutions of nonlinear elliptic problems}, Arch. Rational Mech. Anal. \textbf{114} (1991), no.~1, 79--93. \bibitem{BP05} J.~Byeon and J.~Park; \emph{Singularly perturbed nonlinear elliptic problems on manifolds}, Calc. Var. Partial Differential Equations \textbf{24} (2005), no.~4, 459--477. \bibitem{CC03} A.~Castro and M.~Clapp; \emph{The effect of the domain topology on the number of minimal nodal solutions of an elliptic equation at critical growth in a symmetric domain}, Nonlinearity \textbf{16} (2003), no.~2, 579--590. \bibitem{DY} E.~Dancer and S.~Yan; \emph{Multipeak solutions for a singularly perturbed {N}eumann problem}, Pacific J. Math \textbf{189} (1999), no.~2, 241--262. \bibitem{DMP} E.~Dancer, A.M.~Micheletti, and Angela Pistoia, \emph{Multipeak solutions for some singularly perturbed nonlinear elliptic problems in a {Riemannian} manifold}, to appear on Manus. Math. \bibitem{G} C.~Gui; \emph{Multipeak solutions for a semilinear {N}eumann problem}, Duke Math J. \textbf{84} (1996), no.~3, 739--769. \bibitem{GWW} C.~Gui, J.~Wei, and M.~Winter; \emph{Multiple boundary peak solutions for some singularly perturbed {N}eumann problems}, Ann. Inst. H. Poincar\'{e} Anal. Non Lin\'{e}aire \textbf{17} (2000), no.~1, 47--82. \bibitem{Hta} N.~Hirano; \emph{Multiple existence of solutions for a nonlinear elliptic problem on a {Riemannian} manifold}, Nonlinear Anal., \textbf{70} (2009), no.~2, 671--692. \bibitem{Li} Y.~Y. Li; \emph{On a singularly perturbed equation with {N}eumann boundary condition}, Comm. Partial Differential Equations \textbf{23} (1998), no.~3-4, 487--545. \bibitem{LNT} C.S. Lin, W.M. Ni, and I.~Takagi; \emph{Large amplitude stationary solutions to a chemiotaxis system}, J. Differential Equations \textbf{72} (1988), no.~1, 1--27. \bibitem{MPta} A.~M.~Micheletti and A.~Pistoia; \emph{Nodal solutions for a singularly perturbed nonlinear elliptic problem in a {Riemannian} manifold}, to appear on Adv. Nonlinear Stud. \bibitem{MP2ta} A.~M.~Micheletti and A.~Pistoia; \emph{The role of the scalar curvature in a nonlinear elliptic problem in a {Riemannian} manifold}, Calc. Var. Partial Differential Equation, \textbf{34} (2009), 233--265. \bibitem{NT1} W.~M.~Ni and I.~Takagi; \emph{On the shape of least-energy solutions to a semilinear {N}eumann problem}, Comm. Pure Appl. Math. \textbf{44} (1991), no.~7, 819--851. \bibitem{NT2} W.~M.~Ni and I.~Takagi; \emph{Locating the peaks of least-energy solutions to a semilinear {N}eumann problem}, Duke Math. J. \textbf{70} (1993), no.~2, 247--281. \bibitem{DFW} M.~Del Pino, P.~Felmer, and J.~Wei; \emph{On the role of mean curvature in some singularly perturbed {N}eumann problems}, SIAM J. Math. Anal. \textbf{31} (1999), no.~1, 63--79. \bibitem{Vta} D.~Visetti; \emph{Multiplicity of solutions of a zero-mass nonlinear equation in a {Riemannian} manifold}, J. Differential Equations, \textbf{245} (2008), no.~9, 2397--2439. \bibitem{W1} J.~Wei; \emph{On the boundary spike layer solutions to a singularly perturbed {N}eumann problem}, J. Differential Equations \textbf{134} (1997), no.~1, 104--133. \bibitem{WW} J.~Wei and M.~Winter; \emph{Multipeak solutions for a wide class of singular perturbation problems}, J. London Math. Soc. \textbf{59} (1999), no.~2, 585--606. \end{thebibliography} \end{document}