Mathematical Sciences: Semi-Linear Elliptic Problems

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This project is concerned with the study of the steady states of partial differential equations. In particular, the principal investigator is interested in existence, multiplicity and symmetry questions for solutions of semilinear elliptic problems. The project consists of two parts. In the first part the principal investigator considers semilinear problems involving a parameter and studies the solution structure of the problem for the parameter in a neighborhood of an eigenvalue of the linear part of the problem. Here the principal investigator is mainly interested in semilinear problems whose nonlinear part involves bounded oscillatory terms (Sine Gordon type problems). For such problems the multiplicity of ground states (positive solutions) appears intimately dependent upon the dimension of the independent variable and the geometry of the underlying domain. In fact, very little is known for domains which are not convex. In the second part the principal investigator studies the question of symmetry breaking and the development of interior and boundary layers for positive solution branches of semilinear elliptic problems. Here the principal investigator is mainly interested in problems where the domain is either a sphere or a convex domain with certain reflection symmetries.