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Variational properties of the solutions of semilinear equations under nonresonance conditions. (English) Zbl 1354.34111

Summary: The paper deals with weak solutions of the semilinear operator equation \(Au-cu= J'(u)\) 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 on \(J\), if \(E\) is the energy functional of the equation and \(c\) lies between two eigenvalues \(\lambda_k\) and \(\lambda_{k+1}\), then for any solution \(u\) of the equation, \(E(u)\leq E(u+w)\) for every element w 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 proof of the particular case when \(c=0\). The theory is applicable to elliptic problems.

MSC:

34G20 Nonlinear differential equations in abstract spaces
47J06 Nonlinear ill-posed problems
47J30 Variational methods involving nonlinear operators
35J20 Variational methods for second-order elliptic equations
58E50 Applications of variational problems in infinite-dimensional spaces to the sciences
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