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On global bifurcation for quasilinear elliptic systems on bounded domains. (English) Zbl 1165.35358
Summary: General second order quasilinear elliptic systems with nonlinear boundary conditions on bounded domains are formulated as nonlinear mappings between Sobolev spaces. It is shown that the linearized mapping is a Fredholm operator of index zero. This and the abstract global bifurcation theorem of J. Pejsachowicz and P. J. Rabier [J. Anal. Math. 76, 289–319 (1998; Zbl 0932.47046)] allow us to carry out bifurcation analysis directly on these elliptic systems. At the abstract level, we establish a unilateral global bifurcation result that is needed when studying positive solutions. Finally, we supply two examples of cross-diffusion population model and chemotaxis model to demonstrate how the theory can be applied.

MSC:
35J55Systems of elliptic equations, boundary value problems (MSC2000)
35B32Bifurcation (PDE)
46T20Continuous and differentiable maps in nonlinear functional analysis
58C25Differentiable maps on manifolds (global analysis)
92D25Population dynamics (general)
92C17Cell movement (chemotaxis, etc.)
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