Mathematical Sciences: Nonlinear Elliptic Boundary Value Problems Having Strong Resonance

这项工作将集中于非线性偏微分方程的问题。重点将放在强迫项与第二个变量之比受到某些限制的椭圆方程上。有不同程度的共振，求解这类方程的方法取决于这个程度。对于强共振的情况，在这种情况下，强迫项的积分仍然是有界的，这方面的注意相对较少。工作将集中在由这个条件支配的方程上。将使用拓扑方法来解决存在性问题。Ambrosetti和Rabinowitz的（现在的）经典山口引理不适用于这种情况（Palais-Smale条件无法验证）。现在提出了另一种方法，它可以通过施加边界条件来绕过先前的困难，这将导致某种形式的广义伪梯度映射。然后，这些映射的临界点将表示方程的解。//

This work will focus on problems of nonlinear partial differential equations. Emphasis will be placed on elliptic equations in which the ratio of the forcing term to the second variable are subject to certain restrictions. There are different degrees of resonance and methods for solving such equations depend on this degree. Relatively little attention has been given to the case of strong resonance in which the integral of the forcing term remains bounded. Work will concentrate on equations governed by this condition. A topological approach to the existence problem will be used. The (now) classical mountain pass lemma of Ambrosetti and Rabinowitz does not apply to this setting (the Palais-Smale condition cannot be verified). An alternate approach is now proposed which may bypass earlier difficulties by imposing boundary conditions which will lead to some form of generalized pseudogradient mappings. The critical points of these mappings will then represent solutions of the equations. //