Robert Stańczy

Positive solutions for superlinear elliptic equations

The existence of a positive, radial solution for superlinear elliptic boundary value problem in an exterior domain is proved, by the use of cone expansion fixed point theorem.

Non-Existence and Uniqueness Results for Supercritical Semilinear Elliptic Equations

Non-existence and uniqueness results are proved for several local and non-local supercritical bifurcation problems involving a semilinear elliptic equation depending on a parameter. The domain is star-shaped and such that a Poincaré inequality holds but no other symmetry assumption is required. Uniqueness holds when the bifurcation parameter is in a certain range. Our approach can be seen, in some cases, as an extension of non-existence results for non-trivial solutions. It is based on Rellich–Pohožaev type estimates. Semilinear elliptic equations naturally arise in many applications, for instance in astrophysics, hydrodynamics or thermodynamics. We simplify the proof of earlier results by K. Schmitt and R. Schaaf in the so-called local multiplicative case, extend them to the case of a non-local dependence on the bifurcation parameter and to the additive case, both in local and non-local settings.

Nonlinear Diffusion Models for Self-gravitating Particles

This paper deals with parabolic-elliptic systems of drift-diffusion type modelling gravitational interaction of particles. The main feature is presence of a nonlinear diffusion describing physically relevant density-pressure relations. We study the existence of solutions of the evolution problem, and recall results on the existence of steady states, and the blow up of solutions in cases when drift prevails the diffusion.

On stationary and radially symmetric solutions to some drift–diffusion equations with nonlocal term

In this paper we obtain the existence of a radial solution for some elliptic nonlocal problem with constraints. The problem arises from some drift–diffusion equation modelling among others systems of self-gravitating particles in astrophysics when one looks for its stationary solutions. The model is also related to chemotaxis phenomenon in microbiology and in two dimensional stationary case it appears in the context of hydrodynamics describing 2-D vortices.