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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