Higher order energy expansions for some singularly perturbed Neumann problems

Wei, Juncheng; Winter, Matthias

Comptes Rendus. Mathématique, Tome 337 (2003) no. 1, pp. 37-42.

Nous étudions le problème suivant de perturbations singulières :

$${ϵ}^2\Delta \mathrm{u}-\mathrm{u}+{\mathrm{u}}^p=0\mathrm{dans}\Omega ,\mathrm{u}>0\mathrm{dans}\Omega \mathrm{et}\frac{\partial \mathrm{u}}{\partial \nu }=0\mathrm{sur}\partial \Omega ,$$

où $\Omega$ est un domaine ouvert dans ${ℝ}^N$, ε>0 est une constante petite et p est un exposant souscritique. L'énergie s'écrit alors $J_{ϵ}\left[\mathrm{u}\right]:={\int }_{\Omega }(\frac{{ϵ}^2}{2}{\left|\nabla \mathrm{u}\right|}^2+\frac{1}{2}{u}^2-\frac{1}{\mathrm{p}+1}{u}^{\mathrm{p}+1})\mathrm{d}x$, où $\mathrm{u}\in {\mathrm{H}}^1\left(\Omega \right)$. Ni et Takagi montrent que pour une solution u_ε avec une pic sur la frontière du domaine, on a le développement asymptotique suivant :

$$J_{ϵ}\left[{\mathrm{u}}_{ϵ}\right]={ϵ}^N\left[\frac{1}{2}\mathrm{I}\left[\mathrm{w}\right]-{\mathrm{c}}_1ϵ\mathrm{H}\left({\mathrm{P}}_{ϵ}\right)+\mathrm{o}\left(ϵ\right)\right],$$

où c₁>0 est une constante générique, P_ε est le point unique de maximum local de u_ε et H(P_ε) est la fonction de la courbure moyenne sur la frontière. On établit que :

$$J_{ϵ}\left[{\mathrm{u}}_{ϵ}\right]={ϵ}^N\left[\frac{1}{2}\mathrm{I}\left[\mathrm{w}\right]-{\mathrm{c}}_1ϵ\mathrm{H}\left({\mathrm{P}}_{ϵ}\right)+{ϵ}^2[{\mathrm{c}}_2{\left(\mathrm{H}\left({\mathrm{P}}_{ϵ}\right)\right)}^2+{\mathrm{c}}_3\mathrm{R}\left({\mathrm{P}}_{ϵ}\right)]+\mathrm{o}\left({ϵ}^2\right)\right],$$

où c₂, c₃ sont les constantes génériques et R(P_ε) est la courbure scalaire de Ricci en P_ε. En particulier c₃>0. Nous présentons des applications de ce développement asymptotique.

We consider the following singularly perturbed semilinear elliptic problem:

$${ϵ}^2\Delta \mathrm{u}-\mathrm{u}+{\mathrm{u}}^p=0\mathrm{in}\Omega ,\mathrm{u}>0\mathrm{in}\Omega \mathrm{and}\frac{\partial \mathrm{u}}{\partial \nu }=0\mathrm{on}\partial \Omega ,$$

where $\Omega$ is a bounded smooth domain in ${ℝ}^N$, ε>0 is a small constant and p is a subcritical exponent. Let $J_{ϵ}\left[\mathrm{u}\right]:={\int }_{\Omega }(\frac{{ϵ}^2}{2}{\left|\nabla \mathrm{u}\right|}^2+\frac{1}{2}{u}^2-\frac{1}{\mathrm{p}+1}{u}^{\mathrm{p}+1})\mathrm{d}x$ 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 u_ε, the following asymptotic expansion holds

$$J_{ϵ}\left[{\mathrm{u}}_{ϵ}\right]={ϵ}^N\left[\frac{1}{2}\mathrm{I}\left[\mathrm{w}\right]-{\mathrm{c}}_1ϵ\mathrm{H}\left({\mathrm{P}}_{ϵ}\right)+\mathrm{o}\left(ϵ\right)\right],$$

where c₁>0 is a generic constant, P_ε is the unique local maximum point of u_ε and H(P_ε) is the boundary mean curvature function. In this Note, we obtain the following higher order expansion of J_ε[u_ε]:

$$J_{ϵ}\left[{\mathrm{u}}_{ϵ}\right]={ϵ}^N\left[\frac{1}{2}\mathrm{I}\left[\mathrm{w}\right]-{\mathrm{c}}_1ϵ\mathrm{H}\left({\mathrm{P}}_{ϵ}\right)+{ϵ}^2[{\mathrm{c}}_2{\left(\mathrm{H}\left({\mathrm{P}}_{ϵ}\right)\right)}^2+{\mathrm{c}}_3\mathrm{R}\left({\mathrm{P}}_{ϵ}\right)]+\mathrm{o}\left({ϵ}^2\right)\right],$$

where c₂, c₃ are generic constants and R(P_ε) is the Ricci scalar curvature at P_ε. In particular c₃>0. Applications of this expansion will be given.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[21] Wei, J.; Winter, M. 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