Mathematical Sciences: Nonlinear Elliptic Boundary Value Problems Having Strong Resonance

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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. //