Nonlinear elliptic anisotropic problem with Fourier boundary condition (Q2923067)

The authors obtain an existence and uniqueness result of entropy solution for nonlinear anisotropic elliptic equation with Fourier boundary condition. Precisely NEWLINE\[NEWLINE \begin{cases} -\sum_{i=1}^N D_i a_i(x,D_iu)+|u|^{p_M(x)-2}u=f\quad & \text{in }\Omega, \\ \sum_{i=1}^N a_i(x, D_iu)\nu_i+\lambda u = g\quad & \text{on }\partial \Omega, \end{cases} NEWLINE\]NEWLINE where \(\Omega\subset \mathbb R^N,\) \(N\geq 3\) is a bounded domain with smooth boundary, \(f\in L^1(\Omega),\) \(g\in L^1(\partial \Omega)\) and \(\lambda>0.\)NEWLINENEWLINEThe authors start with proving the existence and uniqueness of a weak solution when the data are bounded and deduce the main result of existence and uniqueness of entropy solution.