×

zbMATH — the first resource for mathematics

Computation of the difference of topology at infinity for Yamabe-type problems on annuli-domains. I, II. (English) Zbl 0966.35043
For \(\varepsilon> 0\) let \(A_\varepsilon= \{x\in\mathbb{R}^n; \varepsilon<|x|< 1/\varepsilon\}\), \(n\geq 3\). The authors consider the nonlinear elliptic problem \[ -\Delta u= u^{(n+2)/n-2},\quad u>0\quad\text{on }A_\varepsilon,\quad u= 0\quad\text{on }\partial A_\varepsilon.\tag{P\(_\varepsilon\)} \] The positive critical points of the functional \(J_\varepsilon\) on \(H^1_0(A_\varepsilon)\), \[ J_\varepsilon(u)= (1/2) \int_{A_\varepsilon}|\nabla u|^2- {n-2\over 2n} \int_{A_\varepsilon}|u|^{2n/n- 2}, \] are solutions of \((\text{P}_\varepsilon)\). This problem is delicate from a variational viewpoint because of the possible existence of critical points at infinity, which are orbits of \(J_\varepsilon\) along which \(J_\varepsilon\) remains bounded, the gradient goes to zero, and the orbits do not converge. To find the solutions of \((\text{P}_\varepsilon)\) by studying the difference of topology between the level sets of \(J_\varepsilon\), it becomes essential to evaluate the topological contribution of the critical points at infinity. In the first part of this work one computes the difference of topology at infinity in the particular case of double blow-up for thin annuli-domains. In the second part one computes it for expanding annuli \((\varepsilon\to 0)\).

MSC:
35J65 Nonlinear boundary value problems for linear elliptic equations
47J30 Variational methods involving nonlinear operators
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
PDF BibTeX Cite
Full Text: DOI
References:
[1] T. Aubin, Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire , J. Math. Pures Appl. (9) 55 (1976), no. 3, 269-296. · Zbl 0336.53033
[2] T. Aubin, Problèmes isopérimétriques et espaces de Sobolev , J. Differential Geom. 11 (1976), no. 4, 573-598. · Zbl 0371.46011
[3] A. Bahri, Critical Points at Infinity in Some Variational Problems , Pitman Res. Notes Math. Ser., vol. 182, Longman Sci. Tech., Harlow, 1989. · Zbl 0676.58021
[4] A. Bahri and J. M. Coron, Vers une théorie des points critiques à l’infini , Bony-Sjöstrand-Meyer seminar, 1984-1985, École Polytech., Palaiseau, 1985, Exp. No. 8, 24. · Zbl 0585.58004
[5] A. Bahri and J. M. Coron, Une théorie des points critiques à l’infini pour l’équation de Yamabe et le problème de Kazdan-Warner , C. R. Acad. Sci. Paris Sér. I Math. 300 (1985), no. 15, 513-516. · Zbl 0585.58005
[6] A. Bahri, Y. Y. Li, and O. Rey, On a variational problem with lack of compactness: the topological effect of the critical points at infinity , Calc. Var. Partial Differential Equations 3 (1995), no. 1, 67-93. · Zbl 0814.35032
[7] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order , Grundlehren Math. Wiss., vol. 224, Springer-Verlag, Berlin, 1977. · Zbl 0361.35003
[8] J. Kazdan, Prescribing the curvature of a Riemannian manifold , CBMS Regional Conf. Ser. in Math., vol. 57, Conf. Board Math. Sci., Washington, D.C., 1985. · Zbl 0561.53048
[9] E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities , Ann. of Math. (2) 118 (1983), no. 2, 349-374. JSTOR: · Zbl 0527.42011
[10] G. Talenti, Best constant in Sobolev inequality , Ann. Mat. Pura Appl. (4) 110 (1976), 353-372. · Zbl 0353.46018
[11] H. Yamabe, On a deformation of Riemannian structures on compact manifolds , Osaka Math. J. 12 (1960), 21-37. · Zbl 0096.37201
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.