Multiplicity of nodal solutions for fourth order equation with clamped beam boundary conditions

In this paper, we study the global structure of nodal solutions of $$\begin{cases} u''''(x)=\lambda h(x)f(u(x)), & 0<x<1, \\ u(0)=u(1)=u'(0)=u'(1)=0,\\ \end{cases}$$ where $\lambda > 0$ is a parameter, $h\in C([0,1],(0, \infty))$, $f\in C(\mathbb{R})$ and $sf(s)>0$ for $|s|>0$. We show the existence of $S$-shaped component of nodal solutions for the above problem. The proof is based on the bifurcation technique

Unilateral Global Bifurcation from Intervals for Fourth-Order Problems and Its Applications

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″″=αx++βx-+ratfx, 0<t<1, x(0)=x(1)=x″(0)=x″(1)=0, where r≠0 is a parameter, a∈C([0,1],(0,∞)), x+=max⁡{x,0}, x-=-min⁡{x,0}, α,β∈C[0,1], and f∈C(R,R), sf(s)>0, for s≠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 f0∈[0,∞) or f∞∈[0,∞], where f0=lims→0f(s)/s and f∞=lims→+∞f(s)/s. We use the unilateral global bifurcation techniques and the approximation of connected components to prove our main results

Unilateral Global Bifurcation for Fourth-Order Problems and Its Applications

We will establish unilateral global bifurcation result for a class of fourth-order problems. Under some natural hypotheses on perturbation function, we show that (λk,0) is a bifurcation point of the above problems and there are two distinct unbounded continua, Ck+ and Ck-, consisting of the bifurcation branch Ck from (μk,0), where μk is the kth eigenvalue of the linear problem corresponding to the above problems. As the applications of the above result, we study the existence of nodal solutions for the following problems: x′′′′+kx′′+lx=rh(t)f(x),  0<t<1, x(0)=x(1)=x′(0)=x′(1)=0, where r∈R is a parameter and k,l are given constants; h(t)∈C([0,1],[0,∞)) with h(t)≢0 on any subinterval of [0,1]; and f:R→R is continuous with sf(s)>0 for s≠0. We give the intervals for the parameter r≠0 which ensure the existence of nodal solutions for the above fourth-order Dirichlet problems if f0∈[0,∞] or f∞∈[0,∞], where f0=lim|s|→0f(s)/s and f∞=lim|s|→+∞f(s)/s. We use unilateral global bifurcation techniques and the approximation of connected components to prove our main results