A Higher-Order Energy Expansion to Two-Dimensional Singularly Neumann Problems


Of concern is the following singularly perturbed semilinear elliptic problem \begin{equation*} \left\{ \begin{array}{c} \mbox{ϵ2Δuu+up=0{\epsilon}^2\Delta u -u+u^p =0 in Ω\Omega}\\ \mbox{u>0u>0 in Ω\Omega and uν=0\frac{\partial u}{\partial \nu}=0 on Ω\partial \Omega}, \end{array} \right. \end{equation*} where Ω\Omega is a bounded domain in RN{\mathbf{R}}^N with smooth boundary Ω\partial \Omega, ϵ>0\epsilon>0 is a small constant and 1<p<(N+2N2)+1< p<\left(\frac{N+2}{N-2}\right)_+. Associated with the above problem is the energy functional JϵJ_{\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 uH1(Ω)u\in H^1(\Omega), where F(u)=0uspdsF(u)=\int_{0}^{u}s^p ds. Ni and Takagi (\cite{nt1}, \cite{nt2}) proved that for a single boundary spike solution uϵu_{\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]I[w] is the energy of the ground state, c1>0c_1 >0 is a generic constant, PϵP_{\epsilon} is the unique local maximum point of uϵu_{\epsilon} and H(Pϵ)H(P_{\epsilon}) is the boundary mean curvature function at PϵΩP_{\epsilon}\in \partial \Omega. Later, Wei and Winter (\cite{ww3}, \cite{ww4}) improved the result and obtained a higher-order expansion of Jϵ[uϵ]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 c2c_2 and c3>0c_3>0 are generic constants and R(Pϵ)R(P_\epsilon) is the scalar curvature at PϵP_\epsilon. However, if N=2N=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 2p<+ 2 \leq p <+\infty. Without loss of generality, we may assume that the boundary near P\in\partial\Om is represented by the graph {x2=ρP(x1)} \{ x_2 = \rho_{P} (x_1) \}. Then we have the following higher order expansion of Jϵ[uϵ]: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)=A1t+A2t2+A3t3P(t)=A_1 t+A_2 t^2+A_3 t^3 is a polynomial, c1c_1, c2c_2, c3c_3 and A1A_1, A2A_2,A3A_3 are generic real constants and S(P_\epsilon)= \rho_{P_\ep}^{(4)} (0). In particular c3<0c_3<0. Some applications of this expansion are given

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