We consider a boundary value problem for a nonlinear differential equation which arises in an option pricing model with transaction costs. We apply the method of upper and lower solutions in order to obtain solutions for the stationary problem. Moreover, we give conditions for the existence of solutions of the general evolution equation.

The one-dimensional stationary full hydrodynamic model for semiconductor devices with non-isentropic pressure is studied. This model consists of the equations for the electron density, electron temperature, and electric field in a bounded domain supplemented with boundary conditions. The existence of a classical subsonic solution with positive particle density and positive temperature is shown in two situations: non-constant and constant heat conductivities. Moreover, we prove uniqueness of a classical solution in the latter case. The existence proofs are based on elliptic estimates, Stampacchia truncation methods, and fixed-point arguments.

A two-point Neumann boundary value problem for a two ion electro-diffusion model reducible to the Painlevé II equation is investigated. The problem is unconventional in that the model equation involves yet-to-be determined boundary values of the solution. In prior work by Thompson, the existence of a solution was established subject to an inequality on the physical parameters. Here, a two-dimensional shooting method is used to show that this restriction may be removed. A practical algorithm for the solution of the boundary value problem is presented in an appendix.