Mathematical Sciences: Semilinear Elliptic Equations and Systems

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Of the five projects outlined in this mathematical research plan, four will focus on semilinear elliptic partial differential equations while the fifth concerns eigenvalue problems for second order ordinary differential equations. The first tasks involve studies of existence and properties of positive solutions of semilinear equations motivated by attempts to prove large deviation results for weighted occupation time of the critical branching Brownian field. These probabilistic questions have been the subject of recent investigations by several researchers. Standard techniques do not apply in this context because of a lack of certain compactness conditions. When the semilinear equations were first introduced, some power of the unknown function appeared as an isolated element of the equation. In the present work, the power of the unknown is multiplied by a continuous function (with zero boundary data). The goal is to obtain conditions on the multiplier which will ensure the existence of positive solutions- especially solutions which are not radial. Related work will seek to quantify the number of nonradial positive solutions of classes of equations in which the number of radial solutions has been established. In addition, work will be done on the problem of maximizing and minimizing the higher order eigenvalues of the time independent Schrodinger equation. The variation takes place as the potential is allowed to change within a class of potentials. The first order case has been resolved for potentials in the classical Lebesgue spaces. Initial efforts will concentrate on the one-dimensional equation.