A note on a nonlinear elliptic problem with a nonlocal coefficient
A note on a nonlinear elliptic problem with a nonlocal coefficient
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DOI：
10.1016/j.jmaa.2015.11.030
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发表时间：
2016-03
影响因子：
1.3
通讯作者：
D. Naimen
中科院分区：
文献类型：
--
作者：
D. Naimen
本文研究了一类非局部非线性椭圆问题（P）{− a（<$Ω| u| 2dx）Δ u= λ u+ up in Ω，u= 0 on <$Ω，其中N≤ 3，Ω <$RN是具有光滑边界的有界区域<$Ω，a是非退化连续函数，p> 1且λ∈ R.我们证明了非局部系数α对（P）解集结构的几种影响。我们首先引入一个尺度观测，并使用一个相关的半线性问题的解集来描述。这使得我们可以得到（P）的解（λ，u）的无界连续。观察到丰富多样的新的分歧和多重性的结果。我们还证明了非局部系数可以诱导不可数多个解决方案，在一个方便的方式。最后，我们从变分的观点给出了一些评论。
In this paper, we investigate a nonlocal and nonlinear elliptic problem,(P){− a (∫ Ω|∇ u| 2 d x) Δ u= λ u+ u p in Ω, u= 0 on∂ Ω, where N≤ 3, Ω⊂ R N is a bounded domain with smooth boundary∂ Ω, a is a nondegenerate continuous function, p> 1 and λ∈ R. We show several effects of the nonlocal coefficient a on the structure of the solution set of (P). We first introduce a scaling observation and describe the solution set by using that of an associated semilinear problem. This allows us to get unbounded continua of solutions (λ, u) of (P). A rich variety of new bifurcation and multiplicity results are observed. We also prove that the nonlocal coefficient can induce up to uncountably many solutions in a convenient way. Lastly, we give some remarks from the variational point of view.