On a class of singular double phase problems with nonnegative weights whose sum can be zero (Q6070308)

In this paper, the double phase problem \[ \left\{ \begin{array}{l} -(\alpha(t)\varphi_p(u')+\beta(t)\varphi_q(u'))' = \lambda h(t) f(u), \quad t\in(0,1), \\ u(0)=0=u_{\kappa,g}(1) \end{array} \right. \] is considered, where \(\lambda>0\), \(\kappa\in\{0,1\}\), \[ u_{\kappa,g}(1):= \left\{ \begin{array}{ll} u(1), & \kappa = 0, \\ u'(1)+g(\lambda,u(1))u(1), & \kappa=1, \end{array} \right. \] and the functions \(\alpha\), \(\beta\), \(f\), \(g\) and \(h\) satisify certain conditions. Existence, nonexistence, and multiplicity results of positive solutions are established. A generalized condition allows \(\alpha(t_0)+\beta(t_0)=0\) at some point \(t_0 \in (0,1)\) and either \(\alpha\equiv0\) or \(\beta\equiv0\) on some subintervals of \((0,1)\) is considered.