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Bifurcation theorems of Rabinowitz type for certain differential operators of the fourth order. (English) Zbl 0781.35004

Summary: This paper was inspired by the works of P. H. Rabinowitz [e.g.: Rocky Mt. J. Math. 3, 161-202 (1973; Zbl 0255.47069); J. Diff. Equations 14, 462-475 (1973; Zbl 0272.35017)]. We study nonlinear eigenvalue problems for some fourth order elliptic partial differential equations with nonlinear perturbation of Rabinowitz type. We show the existence of an unbounded continuum of nontrivial positive solutions bifurcating from \((\mu_ 1,0)\), where \(\mu_ 1\) is the first eigenvalue of the linearization about 0 of the considered problem. We also prove the related theorem for bifurcation from infinity. The results obtained are similar to those proved by Rabinowitz for second order elliptic partial differential equations.

MSC:

35B32 Bifurcations in context of PDEs
35J65 Nonlinear boundary value problems for linear elliptic equations
35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
35J30 Higher-order elliptic equations
35B20 Perturbations in context of PDEs
35G20 Nonlinear higher-order PDEs
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