Entropy solutions for some elliptic anisotropic problems involving variable exponent with Fourier boundary conditions and measure data

This paper is devoted to the study of some nonlinear elliptic anisotropic Fourier boundary-value problems, whose prototype is given by \(\begin{aligned} {\left\{ \begin{array}{ll} - Au + g(x,u,\nabla u) +\delta \vert u\vert ^{p_0(x)-2}u = \mu -\text{ div }\phi (u)\ \hspace{0.2cm} \ \text{ in } \ \ \Omega ,\\ \ \displaystyle \ Bu+\lambda u=h\ \hspace{3.1cm} \text{ on } \ \ \partial \Omega , \end{array}\right. } \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{c}-Au+g(x,u,\mathrm{\nabla }u)+{\delta |u|}^{p_0(x)-2}u=\mu -\text{div}\varphi (u)\text{in}\mathrm{\Omega },\hfill \\ {\displaystyle Bu+\lambda u=h\text{on}\partial \mathrm{\Omega },}\hfill \end{array}\right)\end{array}$$ where the right hand side \(\mu \) $\mu$ belongs to \(L^1(\Omega ) + W^{-1,\vec {p}\,'(x)}(\overline{\Omega })\) $L^1(\mathrm{\Omega })+W^{-1,\overrightarrow{p}{}^{\prime }(x)}(\overline{\mathrm{\Omega }})$ , the operator Au is a Leray-Lions anisotropic operator and \(\phi \in {\mathcal {C}}^0({\mathbb {R}}, {\mathbb {R}}^{N})\) $\varphi \in {\mathcal{C}}^0(\mathbb{R},{\mathbb{R}}^N)$ , the nonlinear term \(g: \Omega \times {\mathbb {R}}\times {\mathbb {R}}^{N}\longrightarrow {\mathbb {R}}\) $g:\mathrm{\Omega }\times \mathbb{R}\times {\mathbb{R}}^N⟶\mathbb{R}$ satisfying some growth condition but no sign condition. We provide an existence result of entropy solutions for this class of anisotropic problems.