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Title:  HigherOrder Energy Expansions and Spike Locations 
Authors:  Winter, M Wei, J 
Keywords:  HigherOrder Energy Expansions, Ricci Curvature 
Publication Date:  2004 
Publisher:  Springer 
Citation:  Calc Var Partial Differential Equations 20 (2004), 403430 
Abstract:  We consider the following singularly perturbed semilinear elliptic problem:
(I)\left\{
\begin{array}{l}
\epsilon^{2} \Delta u  u + f(u)=0 \ \ \mbox{in} \ \Omega, \\
u>0 \ \ \mbox{in} \ \ \Omega \ \ \mbox{and} \
\frac{\partial u}{\partial \nu} =0 \ \mbox{on} \ \partial \Omega,
\end{array}
\right.
where \Om is a bounded domain in R^N with smooth boundary \partial \Om, \ep>0 is a small constant and f is some superlinear but subcritical nonlinearity.
Associated with (I) is the energy functional J_\ep defined by
J_\ep [u]:= \int_\Om \left(\frac{\ep^2}{2} \nabla u^2 + \frac{1}{2} u^2 F(u)\right) dx
\ \ \ \ \ \mbox{for} \ u \in H^1 (\Om),
where F(u)=\int_0^u f(s)ds. Ni and Takagi proved that for a single boundary spike solution u_\ep, the following asymptotic expansion holds:
J_\ep [u_\ep] =\ep^{N} \Bigg[ \frac{1}{2} I[w] c_1 \ep H(P_\ep) + o(\ep)\Bigg],
where c_1>0 is a generic constant, P_\ep is the unique local maximum point of u_\ep and H(P_\ep) is the boundary mean curvature function at P_\ep \in \partial \Om.
In this paper, we obtain a higherorder expansion of J_\ep [u_\ep]:
J_\ep [u_\ep] =\ep^{N} \Bigg[ \frac{1}{2} I[w] c_1 \ep H(P_\ep) + \ep^2 [c_2 (H(P_\ep))^2 + c_3 R (P_\ep)]+ o(\ep^2)\Bigg]
where c_2, c_3 are generic constants
and R(P_\ep) is the Ricci scalar curvature at P_\ep.
In particular c_3 >0. Some applications of this expansion are given. 
URI:  http://bura.brunel.ac.uk/handle/2438/558 
ISSN:  The original publication is available at www.springerlink.com 
Appears in Collections:  School of Information Systems, Computing and Mathematics Research Papers Mathematical Science

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