On some nonlinear degenerate elliptic equations having a lower term in Musielak spaces

In the present paper, we discuss the existence of bounded weak solutions for the degenerate nonlinear elliptic problem $$\begin{aligned} \text{ div }(\Gamma (x,u,\nabla u))+\kappa (x,u,\nabla u)=f, \end{aligned}$$where \(\text{ div }(\Gamma (x,u,\nabla u))\) is a degenerate Leray−Lions operator and defined on non-reflexive Musielak space \(D(A)\subset W_{0}^{1}L_{\varphi }(\Omega )\), such that \(\varphi\) is a Musielak function. The lower term \(\kappa\) satisfies only the growth condition and no sign condition is assumed on it. The source data f are in \(L^{N}(\Omega )\).