Regularity and Existence Results for Degenerate Elliptic Operators

Abstract

In the first section of this paper we study the Hölder-continuity of solutions of the Schrödinger degenerate equation

$$ - \sum\limits_{i,j = 1}^n {\left( {a_{ij} u_{x_i } } \right)_{x_j } + cu = 0,} $$

assuming the potential c belonging to appropriate degenerate Morrey spaces. In the second section we obtain the existence and the uniqueness of the solution of a variational inequality associated to the degenerate operator

$$ Lu = - \sum\limits_{i,j = 1}^n {\left( {a_{ij} \left( x \right)u_{x_i } + d_j u} \right)_{x_j } } + \sum\limits_{i = l}^n {b_i u_{x_i } + cu} $$

assuming the coefficients of the lower terms and the known term belonging to a suitable degenerate Stummel-Kato class. In both cases the weight w, which gives the degeneration, belongs to the Muckenoupt class A ₂.