Regular solutions for nonlinear elliptic equations, with convective terms, in Orlicz spaces

Abstract

We establish some existence and regularity results to the Dirichlet problem, for a class of quasilinear elliptic equations involving a partial differential operator, depending on the gradient of the solution. Our results are formulated in the Orlicz-Sobolev spaces and under general growth conditions on the convection term. The sub-and supersolutions method is a key tool in the proof of the existence results.

A priori estimates for solutions of g-Laplace type problems

“…To end this introduction we want to point out that recently in [4,5,3] some existence results for equations of the type (1.1) were proved by different arguments. In fact, the authors use the method of sub and supersolutions to show the existence of weak solutions to (1.1).…”

Existence and regularity results for nonlinear elliptic equations in Orlicz spaces

“…This new situation requires the use of Young functions and Orlicz spaces. Existence (and regularity) results for problems with A(∇u) = a(|∇u|)∇u can be found in [5,6,10,11,18,23,29]. In [25] the authors deal with an operator depending on the three variables via Young functions of a real variable.…”

“…We stress that Young's functions are also involved in the growth of the convective term f . Similar hypotheses can be found in [10,11,25]. Given the non-variational nature of the problem, we use the method of sub and super solutions, togheter with truncation techniques and the theorem of existence of zeros for monotone operators.…”