Of concern is the
following singularly perturbed semilinear elliptic problem
\begin{equation*}
\left\{ \begin{array}{c}
\mbox{${\epsilon}^2\Delta u -u+u^p =0$ in $\Omega$}\\
\mbox{$u>0$ in $\Omega$ and $\frac{\partial
u}{\partial \nu}=0$ on $\partial \Omega$},
\end{array}
\right.
\end{equation*}
where $\Omega$ is a bounded domain in ${\mathbf{R}}^N$ with smooth
boundary $\partial \Omega$, $\epsilon>0$ is a small constant and
$1< p<\left(\frac{N+2}{N-2}\right)_+$. Associated with the
above problem is the energy functional $J_{\epsilon}$ defined by
\begin{equation*}
J_{\epsilon}[u]:=\int_{\Omega}\left(\frac{\epsilon^2}{2}{|\nabla
u|}^2 +\frac{1}{2}u^2 -F(u)\right)dx
\end{equation*}
for $u\in H^1(\Omega)$, where $F(u)=\int_{0}^{u}s^p ds$.
Ni and Takagi (\cite{nt1}, \cite{nt2}) proved that for a single
boundary spike solution $u_{\epsilon}$, the following asymptotic
expansion holds:
\begin{equation*}
(1) \ \ \ \ \ \ \ \ J_{\epsilon}[u_{\epsilon}]=\epsilon^{N}
\left[\frac{1}{2}I[w]-c_1 \epsilon
H(P_{\epsilon})+o(\epsilon)\right],
\end{equation*}
where $I[w]$ is the energy of the ground state, $c_1 >0$ is a
generic constant, $P_{\epsilon}$ is the unique local maximum point
of $u_{\epsilon}$ and $H(P_{\epsilon})$ is the boundary mean
curvature function at $P_{\epsilon}\in \partial \Omega$. Later,
Wei and Winter (\cite{ww3}, \cite{ww4}) improved the result and
obtained a higher-order expansion of $J_{\epsilon}[u_{\epsilon}]$:
\begin{equation*}
(2) \ \ \ \ \ \ J_{\epsilon}[u_{\epsilon}]=\epsilon^{N}
\left[\frac{1}{2}I[\omega]-c_{1} \epsilon
H(P_{\epsilon})+\epsilon^2 [c_2(H(P_\epsilon))^2 +c_{3}
R(P_\epsilon)]+o(\epsilon^2)\right],
\end{equation*}
where $c_2$ and $c_3>0$ are generic constants and $R(P_\epsilon)$
is the scalar curvature at $P_\epsilon$. However, if $N=2$, the
scalar curvature is always zero. The expansion (2) is no longer sufficient to distinguish spike locations with same mean curvature.
In this paper, we consider
this case and assume that $2 \leq p <+\infty$. Without loss of generality, we may assume that the
boundary near P\in\partial\Om is represented by the graph $\{ x_2 = \rho_{P}
(x_1) \}$. Then we have the following higher order expansion of
$J_\epsilon[u_\epsilon]:$
\begin{equation*}
(3) \ \ \ \ \ J_\epsilon [u_\epsilon]
=\epsilon^N \left[\frac{1}{2}I[w]-c_1
\epsilon H({P_\epsilon})+c_2 \epsilon^2(H({P_\epsilon}))^2 ]
+\epsilon^3
[P(H({P_\epsilon}))+c_3S({P_\epsilon})]+o(\epsilon^3)\right],
\end{equation*}
where H(P_\ep)= \rho_{P_\ep}^{''} (0) is the curvature, $P(t)=A_1 t+A_2 t^2+A_3
t^3$ is a polynomial,
$c_1$, $c_2$, $c_3$ and $A_1$, $A_2$,$A_3$ are generic real
constants and S(P_\epsilon)= \rho_{P_\ep}^{(4)} (0). In
particular $c_3<0$. Some applications of this expansion are given