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Constant-sign and sign-changing solutions of a nonlinear eigenvalue problem involving the \(p\)-Laplacian. (English) Zbl 1212.35109

Summary: For a certain range of the eigenvalue parameter we prove a new multiple and sign-changing solutions theorem. The novelties of our paper are twofold. First, unlike recent papers in the field we do not assume jumping nonlinearities and allow a rather general growth condition on the nonlinearity involved. Second, our approach strongly relies on a combined use of variational and topological arguments (e.g. critical points, mountain-pass theorem, second deformation lemma, variational characterization of the first and second eigenvalue of the \(p\)-Laplacian) on the one hand, and comparison principles for nonlinear differential inequalities, in particular, the existence of extremal constant-sign solutions, on the other hand.

MSC:

35J60 Nonlinear elliptic equations
35J20 Variational methods for second-order elliptic equations
35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
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