Variational properties of the solutions of semilinear equations under nonresonance conditions

The paper deals with weak solutions of the semilinear operator equation \(Au-cu=J^{\prime}\left( u\right))\ in a Hilbert space, where \(A)\ is a positively defined linear operator, \(J)\ is a \(C^{1})\ functional and \(c)\ is not an eigenvalue of \(A)\. Under some assumptions of \(J)\, if \(E)\ is the energy functional of the equation and \(c)\ lies between two eigenvalues \(\leftthreetimes_{k})\ and \(\leftthreetimes_{k+1})\, then for any solution \(u)\ of the equation, \(E\left( u\right) \leq E\left( u+\omega\right))\ for every element \(\omega)\ orthogonal on the first \(k)\ eigenvectors of \(A)\. The proof is based on the application of Ekeland’s variational principle to a suitable modified functional, and differs essentially from the prooof of the particular case when \(c=0)\. The theory is applicable to elliptic problems.