Bounded perturbations of homogeneous quasilinear operators using bifurcations from infinity
Bounded perturbations of homogeneous quasilinear operators using bifurcations from infinity
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使用无穷远分岔的齐次拟线性算子的有界扰动
DOI：
10.1016/j.jde.2003.09.011
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发表时间：
2004
影响因子：
2.4
通讯作者：
P. Takáč
中科院分区：
文献类型：
--
作者：
P. Drábek;Petr Girg;P. Takáč
本文讨论了Dirichlet p-Laplacian Δpin在有界区域Ω <$RN上的非线性问题的存在性结果：[公式：见正文]这里，Δpu = def div（|u| p−2 <$u），其中p∈（1，∞）是一个固定数，h <$h（x，s）是从Ω× R到R的给定函数，λ∈ R代表一个谱参数。我们关注λ接近λ1，包括谐振情况λ=λ1。假设非线性h是Landesman-Lazer型的，但我们也可以处理消失的非线性。我们的渐近方法在某种意义上取代了Lyapunov-Schmidt方法。与半线性情况p=2不同，我们的方法可以处理更一般的非线性，如果p ≥ 2（消失的非线性与非常快的衰减）。
This paper deals with existence results for the following nonlinear problem with the Dirichlet p-Laplacian Δpin a bounded domain Ω⊂ RN: [Formula: see text] Here, Δpu = def div (| ∇ u|p−2∇ u) , where p∈(1,∞) is a fixed number, h≡h(x,s) is a given function from Ω× R into R , and λ∈ R stands for a spectral parameter. We focus on λ close to λ1, including the resonant case λ=λ1. The nonlinearity h is assumed to be of Landesman–Lazer type, but we can deal with vanishing nonlinearities as well. Our asymptotic method substitutes the Lyapunov–Schmidt method in some sense. Unlike in the semilinear case p=2, our method can treat more general nonlinearities if p≠2 (vanishing nonlinearities with very fast decay).