\(H\)-bubbles in a perturbative setting: the finite-dimensional reduction method. (English) Zbl 1079.53012

The authors prove the existence of embedded spheres in \({\mathbb R}^3\) having prescribed mean curvature of the form \(H(u) = H_0 + \varepsilon H_1(u)\) where \(H_0\) is a non-zero constant, \(H_1:{\mathbb R}^3\to{\mathbb R}\) is a \(C^2\) function, and \(\varepsilon\) is of sufficiently small absolute value. Solutions are obtained as critical points of an appropriate energy functional. The finite-dimensional reduction method mentioned in the title refers to the fact that a set of solutions of the unperturbed problem defines a finite dimensional manifold \(Z\) of critical points for the unperturbed energy functional. The manifold \(Z\) is sufficiently well understood so that the authors can construct a \(3\)-dimensional manifold that serves as a natural constraint for the perturbed energy functional.
The final section of the paper contains a non-existence result.


53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
49J10 Existence theories for free problems in two or more independent variables
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[1] A. Ambrosetti and M. Badiale, Variational perturbative methods and bifurcation of bound states from the essential spectrum , Proc. Roy. Soc. Edinburgh Sect. A 128 (1998), 1131-1161. · Zbl 0928.34029
[2] A. Ambrosetti, J. Garcia Azorero, and I. Peral, Elliptic variational problems in \(\mathbb{R}^{N}\) with critical growth , J. Differential Equations 168 (2000), 10-32. · Zbl 0979.35050
[3] A. Ambrosetti and A. Malchiodi, A multiplicity result for the Yamabe problem on \({S}^{n}\) , J. Funct. Anal. 168 (1999), 529-561. · Zbl 0949.53028
[4] T. Aubin, Nonlinear Analysis on Manifolds: Monge-Ampère Equations , Grundlehren Math. Wiss. 252 , Springer, New York, 1982. · Zbl 0512.53044
[5] F. Bethuel and O. Rey, Multiple solutions to the Plateau problem for nonconstant mean curvature , Duke Math. J. 73 (1994), 593-646. · Zbl 0815.53010
[6] H. Brezis and J.-M. Coron, Multiple solutions of H-systems and Rellich’s conjecture , Comm. Pure Appl. Math. 37 (1984), 149-187. · Zbl 0537.49022
[7] -. -. -. -., Convergence of solutions of H-systems or how to blow bubbles , Arch. Rational Mech. Anal. 89 (1985), 21-56. · Zbl 0584.49024
[8] P. Caldiroli, H-bubbles with prescribed large mean curvature , Manuscripta Math. 113 (2004), 125-142. · Zbl 1055.53004
[9] P. Caldiroli and R. Musina, Existence of minimal \(H\)-bubbles , Commun. Contemp. Math. 4 (2002), 177-209. · Zbl 1009.53008
[10] —-, Existence of \(H\)-bubbles in a perturbative setting , to appear in Rev. Mat. Iberoamericana, preprint, · Zbl 1066.53018
[11] S. Chanillo and A. Malchiodi, Asymptotic Morse theory for the equation \(\Delta v= 2v_{x}\wedge v_{y}\) , to appear in Comm. Anal. Geom., · Zbl 1175.35049
[12] Yuxin Ge, Estimations of the best constant involving the \(L^{2}\) norm in Wente’s inequality and compact \(H\)-surfaces into Euclidean space , ESAIM Control Optim. Calc. Var. 3 (1998), 263-300. · Zbl 0903.53003
[13] Yuxin Ge and F. Hélein, A remark on compact \(H\)-surfaces into \(\mathbb{R}^{3}\) , Math. Z. 242 (2002), 241-250. · Zbl 1052.58020
[14] A. Gyemant, “Kapillarität” in Mechanik der flüssigen und gasförmigen körper , Handbuch der Physik 7 , Springer, Berlin, 1927. · JFM 53.0772.19
[15] T. Isobe, On the asymptotic analysis of \(H\)-systems, I: Asymptotic behavior of large solutions , Adv. Differential Equations 6 (2001), 513-546. · Zbl 1142.35345
[16] -. -. -. -., On the asymptotic analysis of \(H\)-systems, II: The construction of large solutions , Adv. Differential Equations 6 (2001), 641-700. · Zbl 1004.35050
[17] R. Musina, The role of the spectrum of the Laplace operator on \(\mathbb{S}^{2}\) in the \(H\)-bubble problem , to appear in J. Anal. Math., preprint, · Zbl 1129.58300
[18] Y. Sasahara, Asymptotic analysis for large solutions of \(H\)-systems , preprint, 1993. · Zbl 0830.35044
[19] K. Steffen, Isoperimetric inequalities and the problem of Plateau , Math. Ann. 222 (1976), 97-144. · Zbl 0345.49024
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