Higher-order energy expansions and spike locations

Abstract.

We consider the following singularly perturbed semilinear elliptic problem: $(I) \left\{ \begin{array}{l} \epsilon^{2} \Delta u - u + f(u)=0 \ \ \mbox{in} \ \Omega, \\ \displaystyle u>0 \ \ \mbox{in} \ \ \Omega \ \ \mbox{and} \ \frac{\partial u}{\partial \nu} =0 \ \mbox{on} \ \partial \Omega, \end{array} \right.$ where $\Omega$ is a bounded domain in R ^N with smooth boundary $\partial \Omega$, $\epsilon > 0$ is a small constant and f is some superlinear but subcritical nonlinearity. Associated with (I) is the energy functional $J_\epsilon$ defined by $J_\epsilon [u]: = \int_\Omega \left(\frac{\epsilon^2}{2} |\nabla u|^2 + \frac{1}{2} u^2- F(u)\right) dx \ \ \ \ \ \mbox{for} \ u \in H^1 (\Omega),$ where $F(u) = \int_0^u f(s)ds$. Ni and Takagi ([29, 30]) proved that for a single boundary spike solution $u_\epsilon$, the following asymptotic expansion holds: $J_\epsilon [u_\epsilon] = \epsilon^{N} \Bigg[ \frac{1}{2} I[w] -c_1 \epsilon H(P_\epsilon) + o(\epsilon)\Bigg],$ where 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 $P_\epsilon \in \partial \Omega$. In this paper, we obtain a higher-order expansion of $J_\epsilon [u_\epsilon]:$ $J_\epsilon [u_\epsilon] = \epsilon^{N} \Bigg[ \frac{1}{2} I[w] -c_1 \epsilon H(P_\epsilon) + \epsilon^2 [c_2 (H(P_\epsilon))^2 + c_3 R (P_\epsilon)]+ o(\epsilon^2)\Bigg]\;,$ where c ₂, c ₃ are generic constants and $R(P_\epsilon)$ is the scalar curvature at $P_\epsilon$. In particular c ₃ > 0. Some applications of this expansion are given.