Higher Order Energy Expansions for Some Singularly Perturbed Neumann Problems

Abstract

We consider the following singularly perturbed semilinear elliptic problem: \epsilon^{2} \Delta u - u + u^p=0 \ \ \mbox{in} \ \Omega, \quad u>0 \ \ \mbox{in} \ \ \Omega \quad \mbox{and} \ \frac{\partial u}{\partial \nu} =0 \ \mbox{on} \ \partial \Omega, where \Om is a bounded smooth domain in R^N, \ep>0 is a small constant and p is a subcritical exponent. Let J_\ep [u]:= \int_\Om (\frac{\ep^2}{2} |\nabla u|^2 + \frac{1}{2} u^2- \frac{1}{p+1} u^{p+1}) dx be its energy functional, where u \in H^1 (\Om). 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. In this paper, we obtain the following higher order 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. Applications of this expansion will be given.