\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 $\mathbb{R}^N$ ($N\geq 3$), $10$ 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 $$\label{sup_{E_k}Ileq} \sup_{E_k}I \leq B_0 k^{\frac{2(p+1)}{(N-m)(p-1)}},$$ while a simple computation shows that $$\label{inf_{S_k}Igeq} \inf_{S_k} I\geq B_1 \lambda_k(\Omega, G)^{\alpha}-B_2,$$ 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 $$\label{kkkkk} \inf_{S_k} I >N_1 \quad \text{for all } k\geq k_0.$$ 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 $$\label{ccccc} C^\ast=\sup_{E_{k^\ast}} I < \infty.$$ 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 $$\label{SSKK} \mathop{\rm dist}\Big(\big(\cup_{k= k_0}^{k^\ast} S_k\big)\cap I^{C^\ast}, \pm P\Big) >0,$$ 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 $$\label{bigcup} \big(\cup_{k= k_0}^{k^\ast} S_k\big)\cap I^{C^\ast}\subset {\mathcal{U}}.$$ 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 \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} 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. \begin{thebibliography}{00} \bibitem{kajikia-2} R. 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