A Class Of Fourth-order Nonlinear Elliptic Partial Differential Equations Derived From The Vibration Model Of An Anisotropic Plate

In this paper, we study the fourth-order elliptic boundary value problems:In it, l is a constant, f?x,y,u?:[0,1]×[0,1]×R?R is a function. The problem is presented as a model of the vibration of a rigid fiber, which is not isotropic.Firstly, We investigate the non-resonance problem. There exist a constant 0<?₀<2?⁴,and let ??[?₀,2?⁴).When the nonlinear term f on the third element has certain monotonicity,we use the upper and lower solution method to get an existence theorem of the solutions of the above equations, and using the monotone iterative technique proving that the solutions of the corresponding iterative sequence convergence to the maximum solution and minimum solution between a pair of upper and lower solutions.Secondly, let ?=2?⁴,the first eigenvalue of the corresponding eigenvalue problem, we study the corresponding problem of the so-called resonance. By using the coincidence degree theory, some existence theorems of solutions are proved under the nonlinear term satisfies the Caratheclory condition and some other conditions.