Estimates and existence theorems for a class of nonlinear degenerate elliptic equations
Vincenzo Esposito · OpenstarTs (Univeristy of Trieste https://www.units.it/) · 1997
Let $\\left\\{ a_{i,j}\\left(x,\\eta\\right)\\right\\} $ be a matrix of bounded Carathéodory functions such that $a_{i,j}\\left(x,\\eta\\right)\\xi_{j},\\xi_{i}\\geq b\\left(\\mid\\eta\\mid\\right)\ u\\left(x\\right)\\mid\\xi\\mid^{2}\\qquad\\forall\\xi\\epsilon\\mathbb{R^{\\textrm{n}}},$ where b: $[0,+\\infty[$$\\rightarrow\\mathbb{R}$ is a positive bounded continuous function and $\ u\\epsilon L^{1},\\frac{1}{\ u}\\epsilon L^{t}$ 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.