Topological degree methods for partial differential operators in generalized Sobolev spaces

The main aim of this paper is to prove, by using the topological degree methods, the existence of solutions for nonlinear elliptic equation Au = f where Au is partial dierential operators of general divergence form.

On some nonlinear parabolic equations with variable exponents and measure data

We prove the existence of renormalized solutions to a class of nonlinear evolution equations, supplemented with initial and Dirichlet condition in the framework of generalized Sobolev spaces. The data are assumed merely integrable.

A note on weak and strong probabilistic solutions for a stochastic quasilinear parabolic equation of generalized polytropic filtration

In this paper, we investigate a class of stochastic quasilinear parabolic problems with nonstandard growth in the functional setting of generalized Sobolev spaces. The deterministic version of the equation was first introduced and studied by Samokhin, as a generalized model for polytropic filtration. We establish an existence result of weak probabilistic solutions when the forcing terms do not satisfy Lipschitz conditions. Under Lipschitzity of the nonlinear external forces, [Formula: see text] and [Formula: see text], we obtain the uniqueness of the weak probabilistic solutions. Combining the uniqueness and the famous Yamada–Watanabe result we prove the existence of the unique strong probabilistic solution.