On the existence of solutions to nonlinear degenerate elliptic equations with measures data (Q1331772)

We prove the existence of solutions of the equation \[ Au = f \text{ in } \Omega, \quad u = 0 \text{ on } \partial \Omega, \tag{1} \] where \(Au = - \text{div} a (x,Du)\), \(f\) is a bounded Radon measure on \(\Omega\), and \(\Omega\) is an open bounded subset of \(\mathbb{R}^ N\) \((N \geq 2)\). Here, the operator \(A\) is not uniformly elliptic, but we only assume that there exists a nonnegative function \(\nu (x)\) defined on \(\overline \Omega\) such that the condition \(a(x,\xi) \xi \geq \nu (x) | \xi |^ P\) holds for a.e. \(x \in \Omega\), \(\forall \xi \in \mathbb{R}^ N\). Our proof is essentially based on considerations of spaces of Sobolev's type with suitable weight factors.