Editor-in-Chief: Ahmad, Shair
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Get Access to Full TextIn this paper we prove a Morse Lemma for degenerate critical points of a function u which satisfies
where , is the unit ball of and f is a smooth nonlinearity. Other results on the nondegeneracy of the critical points and the shape of the level sets are proved.
Keywords: Morse Theory; Elliptic Equations; Level Sets
MSC 2010: 35J15
H. Amann, A note on degree theory for gradient mappings, Proc. Amer. Math. Soc. 85 (1982), no. 4, 591–595. CrossrefGoogle Scholar
J. Arango and O. Perdomo, Morse theory for analytic functions on surfaces, J. Geom. 84 (2005), no. 1–2, 13–22. Google Scholar
N. Aronszajn, A unique continuation theorem for solutions of elliptic partial differential equations or inequalities of second order, J. Math. Pures Appl. (9) 36 (1957), 235–249. Google Scholar
X. Cabré and S. Chanillo, Stable solutions of semilinear elliptic problems in convex domains, Selecta Math. (N. S.) 4 (1998), no. 1, 1–10. CrossrefGoogle Scholar
L. A. Caffarelli and A. Friedman, Partial regularity of the zero-set of solutions of linear and superlinear elliptic equations, J. Differential Equations 60 (1985), no. 3, 420–433. CrossrefGoogle Scholar
C. Conley, Isolated Invariant Sets and the Morse Index, CBMS Reg. Conf. Ser. Math. 38, American Mathematical Society, Providence, 1978. Google Scholar
L. Damascelli, M. Grossi and F. Pacella, Qualitative properties of positive solutions of semilinear elliptic equations in symmetric domains via the maximum principle, Ann. Inst. H. Poincaré Anal. Non Linéaire 16 (1999), no. 5, 631–652. CrossrefGoogle Scholar
E. N. Dancer, Degenerate critical points, homotopy indices and Morse inequalities, J. Reine Angew. Math. 350 (1984), 1–22. Google Scholar
S. Derivière, T. Kaczynski and P.-O. Vallerand-Beaudry, On the decomposition and local degree of multiple saddles, Ann. Sci. Math. Québec 33 (2009), no. 1, 45–62. Google Scholar
G. Dinca and J. Mawhin, Brouwer degree and applications, in preparation. Google Scholar
B. Gidas, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979), no. 3, 209–243. CrossrefGoogle Scholar
F. Gladiali and M. Grossi, Some results for the Gelfand’s problem, Comm. Partial Differential Equations 29 (2004), no. 9–10, 1335–1364. Google Scholar
F. Hamel, N. Nadirashvili and Y. Sire, Convexity of level sets for elliptic problems in convex domains or convex rings: Two counterexamples, Amer. J. Math. 138 (2016), no. 2, 499–527. CrossrefGoogle Scholar
J. Milnor, Morse Theory, Princeton University, Princeton, 1963. Google Scholar
L. E. Payne, On two conjectures in the fixed membrane eigenvalue problem, Z. Angew. Math. Phys. 24 (1973), 721–729. CrossrefGoogle Scholar
Received: 2019-03-07
Accepted: 2019-07-12
Published Online: 2019-09-13
Citation Information: Advanced Nonlinear Studies, ISSN (Online) 2169-0375, ISSN (Print) 1536-1365, DOI: https://doi.org/10.1515/ans-2019-2055.
© 2019 Walter de Gruyter GmbH, Berlin/Boston.
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