非线性项在零点非渐进增长的四阶边值问题单侧全局分歧

Unilateral global bifurcation for fourth-order boundary value problem with non-asymptotic nonlinearity at 0

We present a Dancer-type unilateral global bifurcation result for a class of fourth-order two-point boundary value problem x"+kx"+lx=λh(t)x+g(t, x,λ), 0< t< 1,x(0)=x(1)=x'(0)=x'(1)=0. Under some natural hypotheses on the perturbation function g:(0,1)×R²→R, we show that (λ_k, 0) is a bifurcation point of the above problem. And there are two distinct unbounded continuas, C_k⁺ and C_k^-, consisting of the bifurcation branch C_k from (λ_k, 0), where λ_k is the k-th eigenvalue of the linear problem corresponding to the above problems. As an application of the above result, the global behavior of the components of nodal solutions of the following problem x"+kx"+lx=rh(t)f(x), 0< t< 1, x(0)=x(1)=x'(0)=x'(1)=0 is studied. We obtain the existence of multiple nodal solutions for the problem if f₀=∞, f_∞ ∈ (0, ∞), f₀=f(s)/s, f_∞=f(s)/s.