四阶非线性常微分方程结点解的存在性

Existence of Nodal Solutions for Fourth-Order Nonlinear Ordinary Differential Equations

摘要: 考察四阶边值问题$$\{\begin{array}{l}{u}^{⁗}(t)=\lambda f(t,u(t)),t\in (0,1),\\ u(0)=u(1)=u^{″}(0)=u^{″}(1)=0\end{array}$$结点解的存在性,其中λ > 0是参数,f ∈C([0,1]×ℝ, ℝ)且f(t,s)s > 0,∀t ∈[O, 1]，s≠0,当f 满 足在零处渐近线性增长，在无穷远处超线性增长时，运用分歧理论获得了该问题结点解的存在性，结 果表明，当入的取值位于不同区间时，该问题结点解的个数也会随之改变。

Abstract: We study the existence of nodal solutions for the fourth-order boundary value problem $$\{\begin{array}{l}{u}^{⁗}(t)=\lambda f(t,u(t)),t\in (0,1),\\ u(0)=u(1)=u^{″}(0)=u^{″}(1)=0,\end{array}$$ where λ > 0 is a parameter, f ∈C([0,1]×ℝ, ℝ), and f(t,s)s > 0 for all t ∈[O, 1] and s ≠ 0. Under the assumptions that f is asymptotically linear at the origin and superlinear at infinity, we use bifurcation theory to establish the existence of nodal solutions for the problem. The results show that the number of nodal solutions of the problem varies as λ takes values in different intervals.