Existence and regularity results for some elliptic equations with degenerate coercivity. (English) Zbl 0911.35049

The authors are interested in the study of the following elliptic problem: \[ -\text{div} (a(x,u) \nabla u) = f\quad \text{in }\Omega,\qquad u = 0\quad\text{in }\partial \Omega, \] where \(\Omega\) is a bounded, open subset of \(\mathbb{R}^n\) and \(a(x,s) : \Omega \times \mathbb{R} \to \mathbb{R}\) is a Carathéodory function such that \[ {\alpha \over (1+| s|)^\theta} \leq a(x,s) \leq \beta. \] The authors prove the existence of solutions for the given problem under various assumptions on the function \(f.\)


35J65 Nonlinear boundary value problems for linear elliptic equations
35B65 Smoothness and regularity of solutions to PDEs
35J70 Degenerate elliptic equations