Symmetry of nodal solutions for singularly perturbed elliptic problems on a ball Juncheng WeiMatthias Winter 35J2035J6535J40symmetry of nodal solutionssingular perturbation problemsmethod of moving planesLiapunov-Schmidt reduction In \cite{now1}, it was shown that the following singularly perturbed Dirichlet problem \begin{gather*} \ep^2\Delta u-u+|u|^{p-1}u=0,\quad\mbox{in }\Omega,\\ u=0\quad\mbox{on }\partial\Omega \end{gather*} has a nodal solution $u_{\ep}$ which has the least energy among all nodal solutions. Moreover, it is shown that $u_{\ep}$ has exactly one local maximum point $P_1^{\ep}$ with a positive value and one local minimum point $P_2^{\ep}$ with a negative value, and as $\ep\to0$,$$\varphi(P_1^{\ep},P_2^{\ep})\to\max_{(P_1,P_2)\in\Omega\times\Omega}\varphi(P_1,P_2), $$ where $\varphi(P_1,P_2)=\min(|P_1-P_2|/2,d(P_1,\partial\Omega),d(P_2,\partial\Omega))$. The following question naturally arises: where is the \textit{nodal surface} $\{u_{\ep}(x)=0\}$? In this paper, we give an answer in the case of the unit ball $\Omega=B_1(0)$. In particular, we show that for $\epsilon$ sufficiently small, $P_1^{\ep}$, $P_2^{\ep}$ and the origin must lie on a line. Without loss of generality, we may assume that this line is the $x_1$-axis. Then $u_{\ep}$ must be\textit{even} in $x_j$, $j=2$, $\dots$, $N$, and \textit{odd} in $x_1$. As a consequence, we show that $\{u_{\ep}(x)=0\}=\{x\in B_1(0)\mid x_1=0\}$. Our proof is divided into two steps: first, by using the method of moving planes, we show that $P_1^{\ep}$, $P_2^{\ep}$ and the origin must lie on the $x_1$-axis and $u_{\ep}$ must be even in $x_j$, $j=2$, $\dots$, $N$. Then, using the Liapunov-Schmidt reduction method, we prove the uniqueness of $u_{\ep}$ (which implies the odd symmetry of $u_{\ep}$ in $x_1$). Similar results are also proved for the problem with Neumann boundary conditions. Indiana University Mathematics Journal 2005 text pdf 10.1512/iumj.2005.54.2546 10.1512/iumj.2005.54.2546 en Indiana Univ. Math. J. 54 (2005) 707 - 742 state-of-the-art mathematics http://iumj.org/access/