Existence theorems for some nonlinear fourth order elliptic boundary value problems

Abstract

The general outline of our method is to that of of the second author second order elliptic parabolic equations [S, 71. of our effort is in deriving estimates for linear equations which lead to for the FrCchet derivatives of the relevant nonlinear to which Section 2 devoted, appear to be sharp in to explained below, and feel are independent interest. are stated in and the existence theorems developed in 4. a concluding section, a comparison is made between our results and on the theory of and pseudomonotone operators. There is substantial overlap in classes of to which the results and those obtained by The latter generally allows much greater in the nonlinearity (without priori estimate being available) whereas our results allow for a certain amount of ‘nonmonotonicity’ in the nonlinear terms. Furthermore, here we obtain a constructive algorithm (a variation of Newton’s method) for the solution, and this is a large part of the motivation for this work. In fact, the second author has recently utilized the second order results in [5] to obtain numerical approximations by the finite element method in [2]. We anticipate that the present work can be similarly utilized.