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Entropy solutions for some elliptic anisotropic problems involving variable exponent with Fourier boundary conditions and measure data. (English) Zbl 07848755

Summary: This paper is devoted to the study of some nonlinear elliptic anisotropic Fourier boundary-value problems, whose prototype is given by \[ \begin{cases} - Au + g(x,u,\nabla u) +\delta |u|^{p_0(x)-2}u = & \mu -\operatorname{div} \phi (u) \quad \text{ in } \Omega, \\ Bu+\lambda u=h \quad & \text{ on } \partial \Omega, \end{cases} \] where the right hand side \(\mu\) belongs to \(L^1(\Omega) + W^{-1, {\vec{p}\,}' (x)} (\overline{\Omega})\), the operator \(Au\) is a Leray-Lions anisotropic operator and \(\phi \in \mathcal{C}^0 (\mathbb{R}, \mathbb{R}^N)\), the nonlinear term \(g: \Omega \times \mathbb{R} \times \mathbb{R}^N \longrightarrow \mathbb{R}\) satisfying some growth condition but no sign condition. We provide an existence result of entropy solutions for this class of anisotropic problems.

MSC:

35J25 Boundary value problems for second-order elliptic equations
35J62 Quasilinear elliptic equations
35A01 Existence problems for PDEs: global existence, local existence, non-existence
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