Anisotropic equations in \(L^ 1\) (Q1906546)

Let \(\mu\) be a bounded Radon measure on \(\Omega\). The authors prove existence of a solution of the anisotropic quasilinear Dirichlet problem \[ - \sum^n_{i= 1} {\partial\over \partial x_i} \Biggl(\Biggl|{\partial u\over \partial x_i}\Biggr|^{p_i- 2} {\partial u\over \partial x_i}\Biggr)= \mu \quad \text{in }\Omega,\quad u= 0\quad \text{on }\partial \Omega, \] where \(p_i> 1\), in the anisotropic Sobolev space \(W^{1,q_i}_0= \{v\in W^{1,1}_0\mid \partial v/\partial x_i\in L^{q_i}, i= 1,\dots, n\}\), where \(q_i> 1\) depend on \(p_i\).