Unilateral global bifurcation from intervals for fourth-order problems and its applications (Q2314724)

Summary: We establish a unilateral global bifurcation result from interval for a class of fourth-order problems with nondifferentiable nonlinearity. By applying the above result, we firstly establish the spectrum for a class of half-linear fourth-order eigenvalue problems. Moreover, we also investigate the existence of nodal solutions for the following half-linear fourth-order problems: \(x'''' = \alpha x^+ + \beta x^- + r a \left(t\right) f \left(x\right)\), \(0 < t < 1\), \(x(0) = x(1) = x''(0) = x''(1) = 0\), where \(r \ne 0\) is a parameter, \(a \in C([0,1],(0, \infty))\), \(x^+ = \max \{x, 0 \}\), \(x^- = - \min \{x, 0 \}\), \(\alpha, \beta \in C [0,1]\), and \(f \in C(\mathbb{R}, \mathbb{R})\), \(s f(s) > 0,\) for \(s \ne 0\). We give the intervals for the parameter \(r\) which ensure the existence of nodal solutions for the above fourth-order half-linear problems if \(f_0 \in [0, \infty)\) or \(f_{\infty} \in [0, \infty]\), where \(f_0 = \lim_{\left|s\right| \rightarrow 0} f(s) / s\) and \(f_{\infty} = \lim_{\left|s\right| \rightarrow + \infty} f(s) / s\). We use the unilateral global bifurcation techniques and the approximation of connected components to prove our main results.