Nonlinear elliptic equations with critical Sobolev exponent in nearly starshaped domains

Molle, Riccardo; Passaseo, Donato

Comptes Rendus. Mathématique, Tome 335 (2002) no. 12, pp. 1029-1032.

On montre que, si $\Omega$ satisfait certaines conditions, le problème (1) ci-dessous, pour ε>0 suffisamment petit et k grand, admet des solutions qui pour ε→0 se concentrent et explosent exactement en k points ; les points de concentration s'approchent du bord de $\Omega$ quand k→∞ ; le nombre de solutions est arbitrairement grand pourvu que ε soit suffisamment petit. Parmi les ouverts bornés $\Omega$ qui satisfont ces conditions il y en a aussi de contractibles, qui peuvent même être arbitrairement proches de ouverts étoilés.

Under suitable assumptions on $\Omega$, we show that, for ε>0 small and k large enough, problem (1) below has solutions which concentrate and blow-up as ε→0 at exactly k points; the blowing-up points approach $\partial \Omega$ as k→∞; the number of solutions tends to infinity as ε→0. These assumptions allow $\Omega$ to be contractible and even arbitrarily close to starshaped domains.

[1] Bahri, A.; Coron, J.M. On a nonlinear elliptic equation involving the critical Sobolev exponent: the effect of the topology of the domain, Comm. Pure Appl. Math., Volume 41 (1988), pp. 253-294

[2] Bahri, A.; Li, Y.; Rey, O. On a variational problem with lack of compactness: the topological effect of the critical points at infinity, Calc. Var., Volume 3 (1995) no. 1, pp. 67-93

[3] Brézis, H.; Nirenberg, L. Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., Volume 36 (1983) no. 4, pp. 437-477

[4] Brézis, H.; Peletier, L.A. Asymptotics for elliptic equations involving critical growth (Colombini; Modica; Spagnolo, eds.), P.D.E. and the Calculus of Variations, Birkhäuser, Basel, 1989, pp. 149-192

[5] Dancer, E.N. A note on an equation with critical exponent, Bull. London Math. Soc., Volume 20 (1988) no. 6, pp. 600-602

[6] Dancer, E.N.; Zhang, K. Uniqueness of solutions for some elliptic equations and systems in nearly star-shaped domains, Nonlinear Anal., Volume 41 (2000) no. 5–6, pp. 745-761

[7] Ding, W.Y. Positive solutions of Δu+u^{(n+2)/(n−2)}=0 on contractible domains, J. Partial Differential Equations, Volume 2 (1989) no. 4, pp. 83-88

[8] Han, Z.C. Asymptotic approach to singular solutions for nonlinear elliptic equations involving critical Sobolev exponent, Ann. Inst. H. Poincaré Anal. Non Linéaire, Volume 8 (1991) no. 2, pp. 159-174

[9] R. Molle, D. Passaseo, Concentrating solutions of slightly supercritical elliptic equations in symmetric domains, to appear

[10] R. Molle, D. Passaseo, to appear

[11] R. Molle, A. Pistoia, Concentration phenomena in elliptic problems with critical and supercritical growth, Adv. Differential Equations, to appear

[12] Passaseo, D. Multiplicity of positive solutions of nonlinear elliptic equations with critical Sobolev exponent in some contractible domains, Manuscripta Math., Volume 65 (1989) no. 2, pp. 147-165

[13] Passaseo, D. Multiplicity of positive solutions for the equation $\Delta \mathrm{u}+\lambda \mathrm{u}+{\mathrm{u}}^{2^{*}-1}=0$ in noncontractible domains, Topol. Methods Nonlinear Anal., Volume 2 (1993) no. 2, pp. 343-366

[14] Passaseo, D. Some sufficient conditions for the existence of positive solutions to the equation $-\Delta \mathrm{u}+\mathrm{a}\left(\mathrm{x}\right)\mathrm{u}={\mathrm{u}}^{2^{*}-1}$ in bounded domains, Ann. Inst. H. Poincaré Anal. Non Linéaire, Volume 13 (1996) no. 2, pp. 185-227

[15] Pohožaev, S.I. On the eigenfunctions of the equation Δu+λf(u)=0, Soviet Math. Dokl., Volume 6 (1965), pp. 1408-1411

[16] Rey, O. A multiplicity result for a variational problem with lack of compactness, Nonlinear Anal., Volume 13 (1989) no. 10, pp. 1241-1249

[17] Rey, O. The role of the Green's function in a nonlinear elliptic equation involving the critical Sobolev exponent, J. Funct. Anal., Volume 89 (1990) no. 1, pp. 1-52