This paper treats nonlinear elliptic boundary value problems of the form (1) L[u] = p(x,u) in , on ∂Ω in the Sobolev space , where L is any selfadjoint strongly elliptic linear differential operator of order 2m. Using both topological degree arguments and minimax methods we obtain existence and multiplicity results for the above problem.
We consider three types of semilinear second order PDEs on a cylindrical domain , where is a bounded domain in , . Among these, two are evolution problems of parabolic and hyperbolic types, in which the unbounded direction of is reserved for time , the third type is an elliptic equation with a singled out unbounded variable . We discuss the asymptotic behavior, as , of solutions which are defined and bounded on .
In this paper we want to show how well-known results from the theory of (regular) elliptic boundary value problems, function spaces and interpolation, subordination in the sense of Bochner and Dirichlet forms can be combined and how one can thus get some new aspects in each of these fields.
The Itô integral calculus and analysis on nilpotent Lie grops are used to estimate the number of eigenvalues of the Schrödinger operator for a quantum system with a polynomial magnetic vector potential. An analogue of the Cwikel-Lieb-Rosenblum inequality is proved.
Let be a Schrödinger operator and let be a Schrödinger type operator on , where is a nonnegative potential belonging to certain reverse Hölder class...