A higher‐order energy expansion to two‐dimensional singularly perturbed Neumann problems

Article type: Research Article

Authors: Wei, Juncheng^; | Winter, Matthias | Yeung, Wai‐Kong

Affiliations: Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong E‐mails: {wei; wkyeung}@math.cuhk.edu.hk | Fachbereich Mathematik, Universität Stuttgart, Pfaffenwaldring 57, D‐70569 Stuttgart, Germany E‐mail: [email protected]‐stuttgart.de

Note: [] Corresponding author.

Abstract: Of concern is the following singularly perturbed semilinear elliptic problem \[\left\{\begin{array}{@{}l}{\varepsilon }^{2}\Delta u-u+u^{p}=0\quad \hbox{in}\varOmega ,\\u>0\quad \hbox{in}\varOmega \quad \hbox{and}\quad \dfrac{ \curpartial u}{\curpartial \nu}=0\quad \hbox{on}\curpartial \varOmega ,\end{array}\right.\] where Ω is a bounded domain in RN with smooth boundary \[$\curpartial \varOmega $, ε>0 is a small constant and \[$1<p<(\frac{N+2}{N-2})_{+}$. Associated with the above problem is the energy functional Jε defined by \[J_{\varepsilon }[u]:=\int_{\varOmega }\bigl(\frac{\varepsilon ^{2}}{2}{|\nabla u|}^{2}+\frac{1}{2}u^{2}-F(u)\bigr)\,\mathrm{d}x\] for u∈H1(Ω), where F(u)=∫0usp ds. Ni and Takagi [Comm. Pure Appl. Math. 44 (1991), 819–851; Duke Math. J. 70 (1993), 247–281] proved that for a single boundary spike solution uε, the following asymptotic expansion holds: \[\begin{equation}J_{\varepsilon }[u_{\varepsilon }]=\varepsilon ^{N}\bigl[\frac{1}{2}I[w]-c_{1}\varepsilon H(P_{\varepsilon })+\mathrm{o}(\varepsilon )\bigr],\end{equation} where I[w] is the energy of the ground state, c1>0 is a generic constant, Pε is the unique local maximum point of uε and H(Pε) is the boundary mean curvature function at \[$P_{\varepsilon }\in \curpartial \varOmega $. Later, Wei and Winter [C. R. Acad. Sci. Paris, Ser. I 337 (2003), 37–42; Calc. Var. Partial Differential Equations 20 (2004), 403–430] improved the result and obtained a higher‐order expansion of Jε[uε]: \[\begin{equation}J_{\varepsilon }[u_{\varepsilon }]=\varepsilon ^{N}\bigl[\frac{1}{2}I[\omega]-c_{1}\varepsilon H(P_{\varepsilon })+\varepsilon ^{2}\left[c_{2}\left(H(P_{\varepsilon })\right)^{2}+c_{3}R(P_{\varepsilon })\right]+\mathrm{o}(\varepsilon ^{2})\bigr],\end{equation} where c2 and c3>0 are generic constants and R(Pε) is the scalar curvature at Pε. 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≤p<+∞. Without loss of generality, we may assume that the boundary near \[$P\in \curpartial \varOmega $ is represented by the graph {x2=ρP(x1)}. Then we have the following higher order expansion of Jε[uε]: \[\begin{equation}J_{\varepsilon }[u_{\varepsilon }]=\varepsilon ^{N}\bigl[\frac{1}{2}I[w]-c_{1}\varepsilon H({P_{\varepsilon }})+c_{2}\varepsilon ^{2}\left(H({P_{\varepsilon }})\right)^{2}+\varepsilon ^{3}\left[P\left(H({P_{\varepsilon }})\right)+c_{3}S({P_{\varepsilon }})\right]+\mathrm{o}(\varepsilon ^{3})\bigr],\end{equation} where H(Pε)=ρ″Pε(0) is the curvature, P(t)=A1t+A2t2+A3t3 is a polynomial, c1, c2, c3 and A1, A2, A3 are generic real constants and S(Pε)=ρPε(4)(0). In particular c3<0. Some applications of this expansion are given.

Journal: Asymptotic Analysis, vol. 43, no. 1-2, pp. 75-110, 2005