Partial Differential Equations
Higher order energy expansions for some singularly perturbed Neumann problems
Comptes Rendus. Mathématique, Volume 337 (2003) no. 1, pp. 37-42.

We consider the following singularly perturbed semilinear elliptic problem:

 ${ϵ}^{2}\Delta \mathrm{u}-\mathrm{u}+{\mathrm{u}}^{p}=0\phantom{\rule{10.0pt}{0ex}}\mathrm{in}\phantom{\rule{3.30002pt}{0ex}}\Omega ,\phantom{\rule{10.0pt}{0ex}}\mathrm{u}>0\phantom{\rule{10.0pt}{0ex}}\mathrm{in}\phantom{\rule{3.30002pt}{0ex}}\Omega \phantom{\rule{10.0pt}{0ex}}\mathrm{and}\phantom{\rule{10.0pt}{0ex}}\frac{\partial \mathrm{u}}{\partial \nu }=0\phantom{\rule{10.0pt}{0ex}}\mathrm{on}\phantom{\rule{3.30002pt}{0ex}}\partial \Omega ,$
where $\Omega$ is a bounded smooth domain in ${ℝ}^{N}$, ε>0 is a small constant and p is a subcritical exponent. Let ${J}_{ϵ}\left[\mathrm{u}\right]:={\int }_{\Omega }\left(\frac{{ϵ}^{2}}{2}{|\nabla \mathrm{u}|}^{2}+\frac{1}{2}{u}^{2}-\frac{1}{\mathrm{p}+1}{u}^{\mathrm{p}+1}\right)\phantom{\rule{1.69998pt}{0ex}}\mathrm{d}x$ be its energy functional, where $\mathrm{u}\in {\mathrm{H}}^{1}\left(\Omega \right)$. Ni and Takagi proved that for a single boundary spike solution uε, the following asymptotic expansion holds
 ${J}_{ϵ}\left[{\mathrm{u}}_{ϵ}\right]={ϵ}^{N}\left[\frac{1}{2}\mathrm{I}\left[\mathrm{w}\right]-{\mathrm{c}}_{1}ϵ\mathrm{H}\left({\mathrm{P}}_{ϵ}\right)+\mathrm{o}\left(ϵ\right)\right],$
where c1>0 is a generic constant, Pε is the unique local maximum point of uε and H(Pε) is the boundary mean curvature function. In this Note, we obtain the following higher order expansion of Jε[uε]:
 ${J}_{ϵ}\left[{\mathrm{u}}_{ϵ}\right]={ϵ}^{N}\left[\frac{1}{2}\mathrm{I}\left[\mathrm{w}\right]-{\mathrm{c}}_{1}ϵ\mathrm{H}\left({\mathrm{P}}_{ϵ}\right)+{ϵ}^{2}\left[{\mathrm{c}}_{2}{\left(\mathrm{H}\left({\mathrm{P}}_{ϵ}\right)\right)}^{2}+{\mathrm{c}}_{3}\mathrm{R}\left({\mathrm{P}}_{ϵ}\right)\right]+\mathrm{o}\left({ϵ}^{2}\right)\right],$
where c2, c3 are generic constants and R(Pε) is the Ricci scalar curvature at Pε. In particular c3>0. Applications of this expansion will be given.

Nous étudions le problème suivant de perturbations singulières :

 ${ϵ}^{2}\Delta \mathrm{u}-\mathrm{u}+{\mathrm{u}}^{p}=0\phantom{\rule{10.0pt}{0ex}}\mathrm{dans}\phantom{\rule{3.30002pt}{0ex}}\Omega ,\phantom{\rule{10.0pt}{0ex}}\mathrm{u}>0\phantom{\rule{10.0pt}{0ex}}\mathrm{dans}\phantom{\rule{3.30002pt}{0ex}}\Omega \phantom{\rule{10.0pt}{0ex}}\mathrm{et}\phantom{\rule{10.0pt}{0ex}}\frac{\partial \mathrm{u}}{\partial \nu }=0\phantom{\rule{10.0pt}{0ex}}\mathrm{sur}\phantom{\rule{3.30002pt}{0ex}}\partial \Omega ,$
$\Omega$ est un domaine ouvert dans ${ℝ}^{N}$, ε>0 est une constante petite et p est un exposant souscritique. L'énergie s'écrit alors ${J}_{ϵ}\left[\mathrm{u}\right]:={\int }_{\Omega }\left(\frac{{ϵ}^{2}}{2}{|\nabla \mathrm{u}|}^{2}+\frac{1}{2}{u}^{2}-\frac{1}{\mathrm{p}+1}{u}^{\mathrm{p}+1}\right)\phantom{\rule{1.69998pt}{0ex}}\mathrm{d}x$, où $\mathrm{u}\in {\mathrm{H}}^{1}\left(\Omega \right)$. Ni et Takagi montrent que pour une solution uε avec une pic sur la frontière du domaine, on a le développement asymptotique suivant :
 ${J}_{ϵ}\left[{\mathrm{u}}_{ϵ}\right]={ϵ}^{N}\left[\frac{1}{2}\mathrm{I}\left[\mathrm{w}\right]-{\mathrm{c}}_{1}ϵ\mathrm{H}\left({\mathrm{P}}_{ϵ}\right)+\mathrm{o}\left(ϵ\right)\right],$
c1>0 est une constante générique, Pε est le point unique de maximum local de uε et H(Pε) est la fonction de la courbure moyenne sur la frontière. On établit que :
 ${J}_{ϵ}\left[{\mathrm{u}}_{ϵ}\right]={ϵ}^{N}\left[\frac{1}{2}\mathrm{I}\left[\mathrm{w}\right]-{\mathrm{c}}_{1}ϵ\mathrm{H}\left({\mathrm{P}}_{ϵ}\right)+{ϵ}^{2}\left[{\mathrm{c}}_{2}{\left(\mathrm{H}\left({\mathrm{P}}_{ϵ}\right)\right)}^{2}+{\mathrm{c}}_{3}\mathrm{R}\left({\mathrm{P}}_{ϵ}\right)\right]+\mathrm{o}\left({ϵ}^{2}\right)\right],$
c2, c3 sont les constantes génériques et R(Pε) est la courbure scalaire de Ricci en Pε. En particulier c3>0. Nous présentons des applications de ce développement asymptotique.

Accepted:
Published online:
DOI: 10.1016/S1631-073X(03)00269-3

Juncheng Wei 1; Matthias Winter 2

1 Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong
2 Fachbereich Mathematik, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany
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Juncheng Wei; Matthias Winter. Higher order energy expansions for some singularly perturbed Neumann problems. Comptes Rendus. Mathématique, Volume 337 (2003) no. 1, pp. 37-42. doi : 10.1016/S1631-073X(03)00269-3. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(03)00269-3/

[1] P.F. Adimurthi; S.L. Yadava Interaction between the geometry of the boundary and positive solutions of a semilinear Neumann problem with critical nonlinearity, J. Funct. Anal., Volume 113 (1993) no. 2, pp. 318-350

[2] N. Alikakos; M. Kowalczyk Critical points of a singular perturbation problem via reduced energy and local linking, J. Differential Equations, Volume 159 (1999), pp. 403-426

[3] P. Bates; E.N. Dancer; J. Shi Multi-spike stationary solutions of the Cahn–Hilliard equation in higher-dimension and instability, Adv. Differential Equations, Volume 4 (1999), pp. 1-69

[4] P. Bates; G. Fusco Equilibria with many nuclei for the Cahn–Hilliard equation, J. Differential Equations, Volume 4 (1999), pp. 1-69

[5] E.N. Dancer; S. Yan Multipeak solutions for a singular perturbed Neumann problem, Pacific J. Math., Volume 189 (1999), pp. 241-262

[6] M. del Pino; P. Felmer Spike-layered solutions of singularly perturbed elliptic problems in a degenerate setting, Indiana Univ. Math. J., Volume 48 (1999), pp. 883-898

[7] A. Gierer; H. Meinhardt A theory of biological pattern formation, Kybernetik (Berlin), Volume 12 (1972), pp. 30-39

[8] M. Grossi; A. Pistoia; J. Wei Existence of multipeak solutions for a semilinear Neumann problem via nonsmooth critical point theory, Calc. Var. Partial Differential Equations, Volume 11 (2000), pp. 143-175

[9] C. Gui; J. Wei Multiple interior peak solutions for some singular perturbation problems, J. Differential Equations, Volume 158 (1999), pp. 1-27

[10] C. Gui; J. Wei; M. Winter Multiple boundary peak solutions for some singularly perturbed Neumann problems, Ann. Inst. H. Poincaré, Anal. Non Linéaire, Volume 17 (2000), pp. 47-82

[11] M. Kowalczyk Multiple spike layers in the shadow Gierer–Meinhardt system: existence of equilibria and approximate invariant manifold, Duke Math. J., Volume 98 (1999), pp. 59-111

[12] Y.-Y. Li On a singularly perturbed equation with Neumann boundary condition, Comm. Partial Differential Equations, Volume 23 (1998), pp. 487-545

[13] Y.-Y. Li; L. Nirenberg The Dirichlet problem for singularly perturbed elliptic equations, Comm. Pure Appl. Math., Volume 51 (1998), pp. 1445-1490

[14] C.-S. Lin; W.-M. Ni; I. Takagi Large amplitude stationary solutions to a chemotaxis systems, J. Differential Equations, Volume 72 (1988), pp. 1-27

[15] W.-M. Ni; I. Takagi On the shape of least energy solution to a semilinear Neumann problem, Comm. Pure Appl. Math., Volume 41 (1991), pp. 819-851

[16] W.-M. Ni; I. Takagi Locating the peaks of least energy solutions to a semilinear Neumann problem, Duke Math. J., Volume 70 (1993), pp. 247-281

[17] W.-M. Ni; J. Wei On the location and profile of spike-layer solutions to singularly perturbed semilinear Dirichlet problems, Comm. Pure Appl. Math., Volume 48 (1995), pp. 731-768

[18] W.-M. Ni Diffusion, cross-diffusion, and their spike-layer steady states, Notices Amer. Math. Soc., Volume 45 (1998), pp. 9-18

[19] J. Wei On the construction of single-peaked solutions to a singularly perturbed semilinear Dirichlet problem, J. Differential Equations, Volume 129 (1996), pp. 315-333

[20] J. Wei On the boundary spike layer solutions of singularly perturbed semilinear Neumann problem, J. Diffential Equations, Volume 134 (1997), pp. 104-133

[21] J. Wei; M. Winter Stationary solutions for the Cahn–Hilliard equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, Volume 15 (1998), pp. 459-492

[22] J. Wei, M. Winter, Higher order energy expansions and location of spikes for some singularly perturbed Neumann problems, Submitted

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