Bifurcations of Positive and Negative Continua in Quasilinear Elliptic Eigenvalue Problems期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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Bifurcations of Positive and Negative Continua in Quasilinear Elliptic Eigenvalue Problems

Authors:

Institution:

1. Department of Mathematics, Faculty of Applied Sciences, University of West Bohemia, P.O. Box 314, CZ-30614, Plzeň, Czech Republic
2. Institut für Mathematik, Universit?t Rostock, Universit?tsplatz 1, D-18055, Rostock, Germany

Abstract:

The main result of this work is a Dancer-type bifurcation result for the quasilinear elliptic problem

$\left\{\begin{aligned} -\Delta_p u &= \lambda\vert u \vert^{p-2}u + h\left(x,u(x);\lambda\right)\,\,\hbox{ in }\,\,\Omega;\\ u&= 0\,\,\hbox{on}\,\,\partial\Omega.\\ \end{aligned}\right.$

((P))

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. Submitted: July 28, 2007. Accepted: November 8, 2007.

Keywords:

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