Sign-changing and multiple solutions of the Sturm-Liouville boundary value problem via invariant sets of descending flow (Q555070)

The existence of positive, negative and sign-changing solutions is proved for the second order Sturm-Liouville boundary value problem \[ -(pu')' + qu = f(x,u) \text{ for }x\in [0,1], \] \[ \alpha u'(0) - \beta u(0)= \gamma u'(1) + \sigma u(1) = 0, \] with \(\alpha, \beta, \gamma, \delta \geq 0\), \((\alpha^2 + \beta^2)(\gamma^2+ \delta^2)>0\), \(p, q:[0,1]\to (0,+\infty)\) continuously differentiable functions and \(f:[0,1]\times \mathbb R\to\mathbb R\) continuous and uniformly Lipschitz in \(u\). Several existence results are obtained under different growth conditions on \(f\). The proofs are based on minimax methods and invariant sets of descending flows.