Bifurcation points of a singular boundary-value problem on (0,1)

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• 作者：Charles A. Stuart ^{charles.stuart@epfl.ch" class="auth_mail" title="E-mail the corresponding author}
• 关键词：34B16 ; 34B15 ; 47H30 ; 34C23
• 刊名：Journal of Differential Equations
• 出版年：2016
• 出版时间：5 April 2016
• 年：2016
• 卷：260
• 期：7
• 页码：6267-6321
• 全文大小：633 K

文摘

Following earlier work on some special cases  and  and on the analogous problem in higher dimensions  and , we make a more thorough investigation of the bifurcation points for a nonlinear boundary value problem of the form where, for all λ∈R$\lambda \in \mathbb{R}$ and x∈(0,1)$x\in (0,1)$, f(λ,x,0,0)=0$f(\lambda ,x,0,0)=0$ and so that the formal linearization about a trivial solution u≡0$u\equiv 0$ is Even when f is a smooth function of all its variables, standard bifurcation theory does not apply to the problem and the results differ from the usual conclusions. This is because we deal with the case where the coefficient A   has a critical degeneracy as x→0$x\to 0$ in the sense that It was observed in  and  that if the exponent 2 is replaced by a value less than 2 then classical bifurcation theory can be used to treat the problem. The paper [17] deals with the case f(λ,x,s,t)=λsin⁡s$f(\lambda ,x,s,t)=\lambda \sin s$ whereas [11] covers the more general form f(λ,x,s,t)=λF(s)$f(\lambda ,x,s,t)=\lambda F(s)$. Here we admit a much broader class of nonlinearities and some new phenomena appear. In particular, we encounter situations where bifurcation does not occur at a simple eigenvalue of the linearization lying below the essential spectrum.