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Title:  Symmetry of Nodal Solutions for Singularly Perturbed Elliptic Problems on a Ball 
Authors:  Winter, M Wei, J 
Keywords:  Symmetry of nodal solutions, singular perturbation problems, method of moving planes, LiapunovSchmidt reduction 
Publication Date:  2005 
Publisher:  Indiana Univ Math J 
Citation:  Indiana Univ Math J 54 (2005), 707741 
Abstract:  In [40], it was shown that the following singularly perturbed Dirichlet problem
\ep^2 \Delta u  u+ u^{p1} u=0, \ \mbox{in} \ \Om,\]
\[ u=0 \ \mbox{on} \ \partial \Om
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 \to 0,
\varphi (P_1^\ep, P_2^\ep) \to \max_{ (P_1, P_2) \in \Om \times \Om } \varphi (P_1, P_2),
where \varphi (P_1, P_2)= \min (\frac{P_1P_2}{2}, d(P_1, \partial \Om), d(P_2, \partial \Om)). The following question naturally arises: where is the {\bf nodal surface} \{ u_\ep (x)=0 \}? In this paper, we give an answer in the case of the unit ball \Om=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_1axis.
Then u_\ep must be even in x_j, j=2, ..., N, and odd in x_1.
As a consequence, we show that \{ u_\ep (x)=0 \} = \{ x \in B_1 (0)  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_1axis and u_\ep must be even in x_j, j=2, ..., N. Then,
using the LiapunovSchmidt 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. 
URI:  http://bura.brunel.ac.uk/handle/2438/523 
Appears in Collections:  Mathematical Science Dept of Mathematics Research Papers

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