Differential and Integral Equations

Constant-sign and sign-changing solutions of a nonlinear eigenvalue problem involving the $p$-Laplacian

Siegfried Carl and Dumitru Motreanu

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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.

Article information

Differential Integral Equations Volume 20, Number 3 (2007), 309-324.

First available in Project Euclid: 20 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35J60: Nonlinear elliptic equations
Secondary: 35J20: Variational methods for second-order elliptic equations 35P30: Nonlinear eigenvalue problems, nonlinear spectral theory 47J05: Equations involving nonlinear operators (general) [See also 47H10, 47J25] 47J30: Variational methods [See also 58Exx] 58E05: Abstract critical point theory (Morse theory, Ljusternik-Schnirelman (Lyusternik-Shnirel m an) theory, etc.)


Carl, Siegfried; Motreanu, Dumitru. Constant-sign and sign-changing solutions of a nonlinear eigenvalue problem involving the $p$-Laplacian. Differential Integral Equations 20 (2007), no. 3, 309--324. https://projecteuclid.org/euclid.die/1356039504.

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