Existence of renormalized solutions for some nonlinear elliptic equations in Orlicz spaces

Abstract

In this work, we prove an existence theorem of renormalized solutions for nonlinear elliptic problem of the type $$\begin{aligned} -{\text {div}}\>a(x,u,\nabla u)-\mathop {{\mathrm{div}}}\Phi (x,u)= f \quad \text {in }{\Omega }, \end{aligned}$$where the lower order term $$\Phi $$ verifies the natural growth condition $$\begin{aligned} |\Phi (x,s)|\le \gamma (x)+\overline{M}^{-1}(M(|s|)), \text { with } \gamma \in E_{\overline{M}}(\Omega ). \end{aligned}$$No $$\Delta _{2}$$-condition is needed neither on the N-function M nor on its complementary $$\overline{M}$$.