Existence of renormalized solution of nonlinear elliptic problems with lower order term in Orlicz spaces

Abstract

In this paper, we shall be concerned with the existence of renormalized solution of the following problem, $$\begin{aligned} \left\{ \begin{array}{l} -\text {div}\Big (a(x,u,\nabla u)\Big )-\text {div}(\Phi (x,u))= f \ \ \mathrm{in}\ \Omega ,\\ u=0 \text { on } \partial \Omega , \end{array} \right. \end{aligned}$$ with the second term f belongs to \(L^1(\Omega )\). The growth and the coercivity conditions on the monotone vector field a are prescribed by a N-function M. We assume any restriction on M, therefore we work with Orlicz-Sobolev spaces which are not necessarily reflexive. The lower order term \(\Phi \) is a Caratheodory function which is not coercive.