Abstract.
The main result of this work is a Dancer-type bifurcation result for the quasilinear elliptic problem
Here, Ω is a bounded domain in \({\mathbb{R}}^N (N \geq 1), \Delta_p u\,\, {\mathop = \limits^{\rm def} }\,\, {\rm div}(\mid \nabla u\mid^{p-2}\nabla u)\) denotes the Dirichlet p-Laplacian on \(W^{1,p}_0(\Omega), 1 < p < \infty\), and \(\lambda \in {\mathbb{R}}\) is a spectral parameter. Let μ1 denote the first (smallest) eigenvalue of −Δ p . Under some natural hypotheses on the perturbation function \(h : \Omega \times {\mathbb{R}}\times {\mathbb{R}} \rightarrow {\mathbb{R}}\), we show that the trivial solution \((0, \mu_1) \in E = W^{1,p}_0 (\Omega)\times {\mathbb{R}}\) is a bifurcation point for problem (P) and, moreover, there are two distinct continua, \(\mathcal{Z}^+_{\mu_1}\) and \(\mathcal{Z}^-_{\mu_1}\), consisting of nontrivial solutions \((u,\lambda) \in E\) to problem (P) which bifurcate from the set of trivial solutions at the bifurcation point (0, μ1). The continua \(\mathcal{Z}^+_{\mu_1}\) and \(\mathcal{Z}^-_{\mu_1}\) are either both unbounded in E, or else their intersection \(\mathcal{Z}^+_{\mu_1} \cap \mathcal{Z}^-_{\mu_1}\) contains also a point other than (0, μ1). For the semilinear problem (P) (i.e., for p = 2) this is a classical result due to E. N. Dancer from 1974. We also provide an example of how the union \(\mathcal{Z}^+_{\mu_1} \cap \mathcal{Z}^-_{\mu_1}\) looks like (for p > 2) in an interesting particular case.
Our proofs are based on very precise, local asymptotic analysis for λ near μ1 (for any 1 < p < ∞) which is combined with standard topological degree arguments from global bifurcation theory used in Dancer’s original work.
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Communicated by Rafael D. Benguria.
Submitted: July 28, 2007. Accepted: November 8, 2007.
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Girg, P., Takáč, P. Bifurcations of Positive and Negative Continua in Quasilinear Elliptic Eigenvalue Problems. Ann. Henri Poincaré 9, 275–327 (2008). https://doi.org/10.1007/s00023-008-0356-x
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DOI: https://doi.org/10.1007/s00023-008-0356-x