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This paper was inspired by the works of P. H. Rabinowitz. 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 (μ1, 0), where μ1 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 ([5]–[7]). The methods used are based, in principle, on the results of [1], [5], [6]. Introduction. Let (E, ‖ ‖) be a real Banach space and let L : E → E be a compact linear operator. If there exist μ ∈ R and 0 6= v ∈ E such that v = μLv, then μ is said to be a real characteristic value of L. The set of real characteristic values of L is denoted by r(L). Consider the equation (0.1) u = λLu+H(λ, μ) where H : R× E → E is a compact operator satisfying (0.2) H(λ, u) = o(‖u‖) at u = 0 uniformly on bounded λ intervals. By a solution of (0.1) is meant a pair (λ, u) ∈ R × E satisfying (0.1). In particular, (0.1) has the line of trivial solutions R = {(λ, 0) | λ ∈ R}. If (0.1) has a nontrivial solution in every neighbourhood of (μ, 0), then (μ, 0) is said to be a bifurcation point for (0.1) with respect to the curveR. It is well known that the possible bifurcation points for (0.1) with respect to R lie on {(μ, 0) | μ ∈ r(L)}. Let S denote the closure of the set of nontrivial solutions of (0.1). A component of S is a maximal closed connected subset. The following was shown in [5]: 1991 Mathematics Subject Classification: 35J65, 35P30.

Bifurcation theorems of Rabinowitz type for certain differential operators of the fourth order