A note on a nonlinear elliptic problem with a nonlocal coefficient

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• 作者：Daisuke Naimen¹ ; ² ; ³ ; ^{ds.naimen@gmail.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Nonlocal ; Elliptic ; Bifurcation ; Variational method ; Kirchhoff type
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2016
• 出版时间：15 March 2016
• 年：2016
• 卷：435
• 期：2
• 页码：1710-1737
• 全文大小：608 K

文摘

In this paper, we investigate a nonlocal and nonlinear elliptic problem, where N≤3$N\le 3$, Ω⊂R^N$\mathrm{\Omega }\subset {\mathbb{R}}^N$ is a bounded domain with smooth boundary ∂Ω, a   is a nondegenerate continuous function, p>1$p>1$ and λ∈R$\lambda \in \mathbb{R}$. 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)$(\lambda ,u)$ 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.