summary:We consider the existence of infinitely many solutions to the boundary value problem \begin {gather} \frac {{\rm d}}{{\rm d} t}\Big (\frac {1}{2} _{0}D_{t}^{-\beta }(u'(t)) +\frac {1}{2} _{t}D_{T}^{-\beta }(u'(t))\Big )+\nabla F(t,u(t))=0 \quad \text {\rm a.e.}\ t\in [0,T],\nonumber \\ u(0)=u(T)=0.\nonumber \end {gather} Under more general assumptions on the nonlinearity, we obtain new criteria to guarantee that this boundary value problem has infinitely many solutions in the superquadratic, subquadratic and asymptotically quadratic cases by using the critical point theory.
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