The authors consider the problem (1) in with and , for , being -periodic in and even in and . This problem models a bending beam supported by cables under a load , where describes the fact that cables resist expansion but not compression. The corresponding eigenvalue problem in has eigenvalues and eigenfunctions which form an orthonormal base in the Hilbert space with even in and .
For and the authors establish the existence of positive, negative, or sign-changing solutions in terms of and (note that the nonlinearity crosses the eigenvalue . In -space there is a cone , , , such that for there exist a positive and at least two sign-changing solutions and no negative solution, while on there exist a positive and a sign-changing solution. It is shown that there are three other cones with analogous properties. The proof runs in and its two-dimensional subspace spanned by and , and uses critical point theory, among others.