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Anisotropic equations in $$L^ 1$$. (English) Zbl 0838.35048
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 $$p_i$$.

##### MSC:
 35J70 Degenerate elliptic equations 35J60 Nonlinear elliptic equations 35R05 PDEs with low regular coefficients and/or low regular data
##### Keywords:
bounded Radon measure; existence; anisotropic Sobolev space