Existence and regularity for a class of non-uniformly elliptic equations in two dimensions (Q5933782)

The author considers the following elliptic problem: NEWLINE\[NEWLINE -\text{div}(a(x,u)\nabla u)= f \quad\text{in}\quad \Omega NEWLINE\]NEWLINE NEWLINE\[NEWLINE u=0\quad\text{on}\quad \partial\Omega, NEWLINE\]NEWLINE where \(\Omega\) is bounded open set of \(\mathbb R^2, a(x,s):\Omega\times \mathbb R\to\mathbb R\) is a bounded Caratheodory function, satisfying the following assumption: NEWLINE\[NEWLINE \frac{1}{(1+|s|)^\theta}\leq a(x,s). NEWLINE\]NEWLINE The datum \(f\) is assumed to belong to the space \(L^1(\Omega)\). It is proved existence of entropy solution \(u(x),\) such that \(u\in L^p(\Omega)\) for every \(p\geq 1\) and \(\nabla u\) belongs to the Marcinkievicz space \(M^p(\Omega)\). Additionally, if \(f\) belongs to the Lorentz space \(L(1,1)\), than \(u(x)\) is bounded. It is proved the sharpness of the mentioned regularity result in the scale of Lorentz spaces.