Estimates and existence theorems for a class of nonlinear degenerate elliptic equations

Abstract

Let $\left\{ a_{i,j}\left(x,\eta\right)\right\} $ be a matrix of bounded Carathéodory functions such that $a_{i,j}\left(x,\eta\right)\xi_{j},\xi_{i}\geq b\left(\mid\eta\mid\right)\nu\left(x\right)\mid\xi\mid^{2}\qquad\forall\xi\epsilon\mathbb{R^{\textrm{n}}},$ where b: $[0,+\infty[$$\rightarrow\mathbb{R}$ is a positive bounded continuous function and $\nu\epsilon L^{1},\frac{1}{\nu}\epsilon L^{t}$ with t >1. A priori estimates for solutions of the homogeneous Dirichlet problem related to the equation $-\left(a_{i,j}\left(x,u\right)u_{x_{j}}\right)_{x_{i}}=f$ are proved under various summability assumptions on f. As a consequence, existence theorems are obtained.