Weak solutions for anisotropic nonlinear elliptic problem with variable exponent and measure data (Q2910230)

Let \(\Omega\subset \mathbb{R}^N \;(N\geq 3)\) be a bounded smooth domain and \(\mu\) be a bounded Radon measure.NEWLINENEWLINEIn this paper, the authors study the following anisotropic nonlinear boundary value problem: NEWLINE\[NEWLINE -\sum^N_{i=1} \frac{\partial}{\partial x_i} a_i(x,\frac{\partial u}{\partial x_i}) = \mu \text{ in } \Omega, \;u|_{\partial \Omega} =0, NEWLINE\]NEWLINE where \(a_i(\cdot,\cdot): \Omega\times \mathbb{R} \to \mathbb{R}\) is a Carathéodory function (\(i=1,2,\dots,N\)) and there exists \(C_1>0\) such that NEWLINE\[NEWLINE|a_i(x,\xi)| \leq C_1(1+|\xi|^{p_i(x)-1}) \text{ for all } \xi \in \mathbb{R} \text{ and a.e. } x\in \Omega, \;i=1,2,\dots, N.NEWLINE\]NEWLINE Under some further conditions on \(a_i(x,\xi)\) and \(p_i(x)\), the existence of a weak solution for the above nonlinear elliptic problem is proved in an anisotropic variable exponent Sobolev space.