Research On Partial Differential Equations Arisen From The Vibration Model Of Rigid Fiber Woven Materials

In this paper,we discuss the boundary value problem for a fourth-order equationwhere f(x,y,u)?C([0,1]×[0,1]×R).This problem was arisen from the vibration model of perpendicular rigid fiber woven plates with an anisotropic grid.This grid is hinged at the boundary of the square.Under some given conditions about f(x,y,u) the above problem has the existence theorems of weak solutions by the critical theory.Firstly,we introduce the background,origin and significance of this equation and analyse the domestic and foreign research present situations.Then,we give the preparing knowledge and necessary lemmas for this paper.Afterwards,we study that the above problem has the existence of the nontrivial solutions,when f(x,y,u)satisfies Ambrosetti-Rabinowitz conditions.The last,we study that the above problem has the existence of the nontrivial solutions,when f(x,y,u)does not satisfy Ambrosetti-Rabinowitz conditions with asymptotical linearity and non-resonance,the first eigenvalue resonance,other eigenvalues resonance and super-linear growth respectively.