Existence of entropy solutions for some anisotropic quasilinear elliptic unilateral problems
Tóm tắt
In this work, we consider the following quasilinear elliptic unilateral equations of the type
$$\begin{aligned} -\sum _{i=1}^{N}\frac{\partial }{\partial x_{i}}a_{i}(x,u,\nabla u) = \mu - \text{ div } \phi (u)\quad \text{ in } \Omega . \end{aligned}$$
In the anisotropic Sobolev space, we prove the existence of entropy solutions for our unilateral problem, where
$$\mu = f-\text{ div } F$$
belongs to
$$L^{1}(\Omega ) + W^{-1,\mathbf {p'}}(\Omega )$$
and
$$\phi (\cdot ) \in C^{0}(\mathbb {R},\mathbb {R}^{N}).$$
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