Entropy solution for a nonlinear elliptic problem with lower order term in Musielak–Orlicz spaces

Abstract

In this paper, we shall be concerned with the existence result of the following problem, 0.1 $$\begin{aligned} \left\{ \begin{array}{l} -\text {div}\left( a(x,u,\nabla u)\right) -\text {div}(\Phi (x,u))= f \ \ \mathrm{in}\ \Omega ,\\ u=0 \text { on } \partial \Omega , \end{array} \right. \end{aligned}$$ 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 generalized N-function M. We assume any restriction on M, therefore we work with Musielak–Orlicz spaces which are not necessarily reflexive. The lower order term $$\Phi $$ is a Caratheodory function satisfying only a growth condition.