Global structure for a fourth-order boundary value problem with sign-changing weight

We study the fourth-order boundary value problem with a sign-changing weight function: u ⁗ = λ m ( t ) u + f 1 ( t , u , u ′ , u ″ , u ‴ , λ ) + f 2 ( t , u , u ′ , u ″ , u ‴ , λ ) , t ∈ ( 0 , 1 ) , u ( 0 ) = u ( 1 ) = u ″ ( 0 ) = u ″ ( 1 ) = 0 , $$\left\{\begin{array}{ll} u''''=\lambda m(t)u+f_1(t,u,u',u'',u''',\lambda)+f_2(t,u,u',u'',u''',\lambda),\qquad t\in(0,1),\\[1.3ex] u(0)=u(1)=u''(0)=u''(1)=0, \end{array}\right.$$ where λ ∈ ℝ is a parameter, f 1, f 2 ∈ C([0, 1] × ℝ5, ℝ), f 1 is not differentiable at the origin and infinity. Under some suitable conditions on nonlinear terms, we prove the existence of unbounded continua of positive and negative solutions of this problem which bifurcating from intervals of the line of trivial solutions or from infinity, respectively.