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On the Morse critical groups for indefinite sublinear elliptic problems. (English) Zbl 1087.35040
The author studies a parameter dependent semilinear Dirichlet problem with weight by methods of infinite dimensional Morse theory. Specifically, it is shown that, if a parameter in the problem is not too large, the Morse critical groups at zero for the associated energy functional are trivial. This implies the existence of a nontrivial solution when some nonlinear term in the problem is asymptotically linear. Then the bifurcation of small solutions is obtained under the assumption that a parameter related to the weight is sufficiently small.

35J60Nonlinear elliptic equations
35B32Bifurcation (PDE)
35J20Second order elliptic equations, variational methods
47J30Variational methods (nonlinear operator equations)
58E05Abstract critical point theory
Full Text: DOI
[1] Adams, D. R.; Hedberg, L. I.: Function spaces and potential theory. (1996) · Zbl 0834.46021
[2] Alama, S.: Semilinear elliptic equations with sublinear indefinite nonlinearities. Adv. differential equations 4, 813-842 (1999) · Zbl 0952.35052
[3] Alama, S.; Del Pino, M.: Solutions of elliptic equations with indefinite nonlinearities via Morse theory and linking. Ann. inst. H. Poincarè anal. Non lineaire 13, 95-115 (1996) · Zbl 0851.35037
[4] Ambrosetti, A.; Brezis, H.; Cerami, G.: Combined effect of concave and convex nonlinearities in some elliptic problems. J. funct. Anal. 122, 519-543 (1994) · Zbl 0805.35028
[5] Ambrosetti, A.; Azorero, J. Garsia; Alonso, I. Peral: Multiplicity results for some nonlinear elliptic equations. J. funct. Anal. 137, 219-242 (1996) · Zbl 0852.35045
[6] Bandle, C.; Pozio, M. A.; Tesei, A.: The asymptotic behavior of the solutions of degenerate parabolic equations. Trans. amer. Math. soc. 303, 487-501 (1987) · Zbl 0633.35041
[7] Bartsch, T.; Li, S.: Critical point theory for asymptotically quadratic functionals and applications to problems with resonance. Nonlinear anal. 28, 419-441 (1997) · Zbl 0872.58018
[8] Bartsch, T.; Wang, Z. -Q.: On the existence of sign changing solutions for semilinear Dirichlet problems. Topology methods nonlinear anal. 7, 115-131 (1996) · Zbl 0903.58004
[9] Chang, K. C.: Infinite dimensional Morse theory and multiple solution problems. (1993) · Zbl 0779.58005
[10] Chang, K. C.; Ghoussoub, N.: The Conley index and the critical groups via an extension of Gromoll-Meyer theory. Topology methods nonlinear anal. 7, 77-93 (1996) · Zbl 0898.58006
[11] Dancer, E. N.; Du, Y.: The generalized Conley index and multiple solutions of semilinear elliptic problems. Abstracts appl. Anal. 1, 103-135 (1996) · Zbl 0933.35069
[12] Fuglede, B.: Continuous domain dependence of the eigenvalues of the Dirichlet Laplacian and related operators in Hilbert space. J. funct. Anal. 167, 183-200 (1999) · Zbl 0948.47047
[13] Jin, Z.: Multiple solutions for a class of semilinear elliptic equations. Proc. amer. Math. soc. 125, 3659-3667 (1997) · Zbl 0885.35036
[14] Li, S.; Wang, Z. -Q.: Mountain pass theorem in order intervals and multiple solutions for semilinear elliptic Dirichlet problems. J. anal. Math. 81, 373-396 (2000) · Zbl 0962.35065
[15] Mawhin, J.; Willem, M.: Critical point theory and Hamiltonian systems. (1989) · Zbl 0676.58017
[16] Mercuri, F.; Palmieri, G.: Morse theory with low differentiability. Boll. unione mat. Ital. B 1, No. 7, 621-631 (1987) · Zbl 0633.58014
[17] Moroz, V.: Solutions of superlinear at zero elliptic equations via Morse theory. Topology methods nonlinear anal. 10, 387-398 (1997) · Zbl 0919.35048
[18] Namba, T.: Density-dependent dispersal and spatial distribution of a population. J. theoret. Biol. 86, 351-363 (1980)
[19] Ouyang, T.: On the positive solutions of semilinear equations ${\Delta} u+{\lambda}$ u+hup=0 on compact manifolds. II. Indiana univ. Math. J. 40, 1083-1141 (1991) · Zbl 0773.35020
[20] Perera, K.: Critical groups of pairs of critical points produced by linking subsets. J. differential equations 140, 142-160 (1997) · Zbl 0889.58025
[21] Pozio, M. A.; Tesei, A.: Support properties of solutions for a class of degenerate parabolic problems. Comm. partial differential equations 12, 47-75 (1987) · Zbl 0629.35071
[22] Spanier, E. H.: Algebraic topology. (1966) · Zbl 0145.43303
[23] Wang, Z. -Q.: Nonlinear boundary values problems with concave nonlinearities near the origin. Nodea nonlinear differential equations appl. 8, 15-33 (2001)