Mathematical Sciences: On Nonlinear Elliptic Equations

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Work on this project extends earlier contributions to the theory of semilinear elliptic partial differential equations. Recent research has established that multiple solutions can exist when the equations are defined on unbounded domains. One of the issues to be considered in this project is to clarify the conditions one must impose on the equations to guarantee the existence of any solutions. Simple examples illustrate how an equation may have only trivial solutions. A second line of investigation will focus on second order fully nonlinear elliptic equations. This is an area of considerable activity where existence and uniqueness results have been established for equations on compact Riemannian manifolds with nonnegative sectional curvature. It is not clear whether the nonnegativity is necessary. Work will be done in an effort to explain the role of the sectional curvature and to consider the related complex equations as a longer-range goal.