×

zbMATH — the first resource for mathematics

Critical points of embeddings of \(H_ 0^{1,n}\) into Orlicz spaces. (English) Zbl 0664.35022
From the author’s abstract: For a domain \(\Omega \subset {\mathbb{R}}^ n\) embeddings \(u\to \exp (\alpha (| u| /\| u\|_{1,n})^{n/n-1})\) of \(H_ 0^{1,n}(\Omega)\) into Orlicz spaces are considered. At the critical exponent \(\alpha =\alpha_ n\) a loss of compactness reminiscent of the Yamabe problem is encountered; however, by a result of Carleson and Chang, if \(\Omega\) is a ball the best constant for the above embedding is attained.
In dimension \(n=2\), the author identifies the “limiting problem” responsible for the lack of compactness at the critical exponent \(\alpha_ 2=4\pi\) in the radially symmetric case and establishes the existence of extremal functions also for nonsymmetric domains \(\Omega\). Moreover, he establishes the existence of two “branches” of critical points of this embedding beyond the critical exponent \(\alpha_ 2=4\pi\). References include 16 items.
Reviewer: J.E.Bouillet

MSC:
35J60 Nonlinear elliptic equations
58E15 Variational problems concerning extremal problems in several variables; Yang-Mills functionals
35J20 Variational methods for second-order elliptic equations
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML
References:
[1] Brezis, H.; Coron, J.-M., Convergence of solutions to H-systems or how to blow bubbles, Arch. Rat. Mech. Anal., Vol. 89, 21-56, (1985) · Zbl 0584.49024
[2] L. Carleson and S. Y. A. Chang, On the Existence of an Extremal Function for an inequality of J. Moser, Bull. des Sciences (to appear). · Zbl 0619.58013
[3] Donaldson, S. K., An application of gauge theory to four‐dimensional topology, J. Diff. Eq., Vol. 18, 279-315, (1983) · Zbl 0507.57010
[4] P.-L. Lions, The Concentration-Compactness Principle in the Calculus of Variations. The Limit Case, part 1, Riv. Mat. Iberoamericana, 1985. · Zbl 0704.49005
[5] Monahan, J. P., Numerical solution of a non-linear boundary-value problem, (1971), Princeton Univ., Thesis
[6] Moser, J., A sharp form of an inequality by N. Trudinger, Ind. Univ. Math. J., Vol. 20, 1077-1091, (1971) · Zbl 0213.13001
[7] Palais, R. S., Critical point theory and the minimax principle, Proc. Symp. Pure Math., Vol. XV, 185-212, (1970) · Zbl 0212.28902
[8] Polya, G.; Szegö, G., Isoperimetric inequalities in mathematical physics, (1951), Princeton Univ. Press · Zbl 0044.38301
[9] Sacks, J.; Uhlenbeck, K., The existence of minimal immersions of 2-spheres, Ann. Math., Vol. 113, 1-24, (1981) · Zbl 0462.58014
[10] Sedlacek, S., A direct method for minimizing the yang‐mills functional over 4‐ manifolds, Comm. Math. Phys., Vol. 86, 515-528, (1982) · Zbl 0506.53016
[11] Struwe, M., A global compactness result for elliptic boundary value problems involving limiting non-linearities, Math. Z., Vol. 187, 511-517, (1984) · Zbl 0535.35025
[12] Struwe, M., Large H-surfaces via the mountain-pass-lemma, Math. Ann., Vol. 270, 441-459, (1985) · Zbl 0582.58010
[13] Struwe, M., The existence of surfaces of constant mean curvative with free boundaries, Acta Math., Vol. 160, 19-64, (1988) · Zbl 0646.53005
[14] Taubes, C. H., Path connected Yang-Mills moduli spaces, J. Diff. Geom., Vol. 19, 337-392, (1984) · Zbl 0551.53040
[15] Trudinger, N. S., On imbeddings into Orlicz spaces and some applications, J. Math. Mech., Vol. 17, 473-484, (1967) · Zbl 0163.36402
[16] Wente, H. C., Large solutions to the volume constrained plateau problem, Arch. Rat. Mech. Anal., vol. 75, 59-77, (1980) · Zbl 0473.49029
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.