Symmetric equilibria for a beam with a nonlinear elastic foundation.

We study the existence of symmetric solutions of a nonlinear fourth order O.D.E. with nonlinear boundary conditions arising in the theory of elastic beams. Variational methods are used, namely, duality, minimization and mountain pass. 1 – Introduction This paper is concerned with the study of symmetric solutions of the boundary value problem (1.1) u + g(x, u) = 0 , (1.2) u(0) = u(1) = 0 , (1.3) u(0) = −f(u(0)) and u(1) = f(u(1)) , where g : [0, 1]× IR→ IR and f : IR→ IR are real continuous functions with (1.4) f(s) = 0 iff s = 0 . In order to look for symmetric solutions, i.e. solutions such that u(x) = u(1− x) , the function g will be supposed to satisfy (1.5) g(x, u) = g(1− x, u) Received : July 24, 1992; Revised : October 1, 1992. * Supported by Fundação Calouste Gulbenkian/Portugal. ** Partially supported by CAPES/Brazil. 376 M.R. GROSSINHO and T.F. MA for all u ∈ IR and x ∈ [0, 1]. Variational methods will be used throughout. More precisely we shall work in the Hilbert subspace of the Sobolev space H2(0, 1) that consists of all symmetric functions, that is, H s (0, 1) = { u ∈ H(0, 1) : u(x) = u(1− x) } . In an analogous way we denote by W s (0, 1) and L p s(0, 1) the subspaces of symmetric functions that belong to W(0, 1) and L(0, 1), respectively. Thus problem (1.1)–(1.3) becomes

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