×

Multibumps analysis in dimension 2: quantification of blow-up levels. (English) Zbl 1281.35045

Summary: In this article, we describe the asymptotic behavior of sequences of solutions to some semilinear elliptic equations with critical exponential growth in planar domains. We prove, in particular, a result analogous to that of M. Struwe [Math. Z. 187, 511–517 (1984; Zbl 0535.35025)] in higher dimensions and extend the two-dimensional result of Adimurthi and M. Struwe [J. Funct. Anal. 175, No. 1, 125–167 (2000; Zbl 0956.35045)] to arbitrary energies. We thus answer a question explicitly asked in this last article.

MSC:

35J61 Semilinear elliptic equations
35B45 A priori estimates in context of PDEs
35J25 Boundary value problems for second-order elliptic equations
47J30 Variational methods involving nonlinear operators
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Adimurthi and O. Druet, Blow-up analysis in dimension \(2\) and a sharp form of Trudinger-Moser inequality , Comm. Partial Differential Equations 29 (2004), 295–322. · Zbl 1076.46022 · doi:10.1081/PDE-120028854
[2] Adimurthi and S. Prashanth, Failure of Palais-Smale condition and blow-up analysis for the critical exponent problem in \(\mathbb R^2\) , Proc. Indian Acad. Sci. Math. Sci. 107 (1997), 283–317. · Zbl 0905.35031 · doi:10.1007/BF02867260
[3] Adimurthi and M. Struwe, Global compactness properties of semilinear elliptic equations with critical exponential growth , J. Funct. Anal. 175 (2000), 125–167. · Zbl 0956.35045 · doi:10.1006/jfan.2000.3602
[4] W. X. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations , Duke Math. J. 63 (1991), 615–622. · Zbl 0768.35025 · doi:10.1215/S0012-7094-91-06325-8
[5] -, Qualitative properties of solutions to some nonlinear elliptic equations in \(\mathbb R^2\) , Duke Math. J. 71 (1993), 427–439. · Zbl 0923.35055 · doi:10.1215/S0012-7094-93-07117-7
[6] K. S. Chou and T. Y.-H. Wan, Asymptotic radial symmetry for solutions of \(\Delta u +e^u=0\) in a punctured disc, Pacific J. Math. 163 (1994), 269–276. · Zbl 0794.35049 · doi:10.2140/pjm.1994.163.269
[7] O. Druet, E. Hebey, and F. Robert, Blow-Up Theory for Elliptic PDEs in Riemannian Geometry , Math. Notes 45 , Princeton Univ. Press, Princeton, 2004. · Zbl 1059.58017
[8] O. Druet and F. Robert, Bubbling phenomena for fourth-order four-dimensional PDEs with exponential growth , Proc. Amer. Math. Soc. 134 (2006), 897–908. · Zbl 1083.58018 · doi:10.1090/S0002-9939-05-08330-9
[9] J. Moser, A sharp form of an inequality by N. Trudinger , Indiana Univ. Math. J. 20 (1970/71), 1077–1092. · Zbl 0203.43701 · doi:10.1512/iumj.1971.20.20101
[10] M. Nolasco and G. Tarantello, Double vortex condensates in the Chern-Simons-Higgs theory , Calc. Var. Partial Differential Equations 9 (1999), 31–94. · Zbl 0951.58030 · doi:10.1007/s005260050132
[11] J. Prajapat and G. Tarantello, On a class of elliptic problems in \(\mathbb R^2\) : Symmetry and uniqueness results, Proc. Roy. Soc. Edinburgh Sect. A 131 (2001), 967–985. · Zbl 1009.35018 · doi:10.1017/S0308210500001219
[12] M. Struwe, A global compactness result for elliptic boundary value problems involving limiting nonlinearities , Math. Z. 187 (1984), 511–517. · Zbl 0535.35025 · doi:10.1007/BF01174186
[13] N. S. Trudinger, On embedding into Orlicz spaces and some applications , J. Math. Mech. 17 (1967), 473–483. · Zbl 0163.36402
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.