On a class of degenerate elliptic equations in weighted Hölder spaces. (English) Zbl 1150.35420

The paper is devoted to the study of the Dirichlet problem for the degenerate elliptic equation \[ P_0\Delta u+\gamma (\nabla P_0,\nabla u)=f(x,u,\nabla u) \text{ in } \Omega ,\quad u=0 \text{ on } \partial \Omega , \] where \(\gamma \geq 0\) is a given parameter, \(\Omega \subset \mathbb R^n\) is an annular region, the given function \(P_0(x)\) is such that \(| \nabla P_0| +P_0\geq \varepsilon >0\) in \(\overline \Omega \), and \(P_0=0\) on the outer boundary of \(\Omega \). The existence of a unique classical solution is proved via Schauder estimates in Hölder spaces.


35J70 Degenerate elliptic equations
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35B65 Smoothness and regularity of solutions to PDEs