The authors study a variational problem for real valued functions defined on an infinite semiaxis of the line. They seek a “minimal solution” to the following problem: minimize the functional
Here denotes the Sobolev space of functions possessing a locally integrable second derivative, and is a smooth function satisfying and is a given value. The organization of the paper is as follows: In Section 1 the connection of the variational problem (1),(2) with the equilibrium problem of a long slender bar of polymetric material under tension is discussed. In Section 2 a fixed endpoint variational problem is analyzed. In Section 3 a criterion is given for the solution of (1),(2) to be minimal. Section 4 gives the proof the existence of a minimal energy solution. The Section 5 is containing the main result of the paper: there always exists a periodic minimal solution of (1),(2). In Section 7 an interesting analytic result is established.