On a nonsmooth fourth order boundary value problem

Existence of solutions of the following boundary value problem is investigated by a variational approach. uiv−au″+bu∈∂¯F(t,u),(−u‴(0)+au′(0)u‴(1)−au′(1)u″(0)−u″(1))∈∂j(u(0)u(1)u′(0)u′(1)). Here, F(t,ξ):(0,1)×R→R is a Carathéodory mapping, locally Lipschitz with respect to the second variable, and ∂¯F(t,ξ) denotes the generalized Clarke gradient of F(t,ξ) with respect to ξ, while j is assumed to be a proper, convex, lower semicontinuous function whose subdifferential is denoted by ∂j. This problem is a model for real beam/shell applications.