Weak Solutions of Some Quasilinear Elliptic Equations with Data Measures

Source abstract

Existence and uniqueness of weak solutions for some quasilinear elliptic equations with data measures and arbitrary growth with respect to the gradient are studied. Usual techniques based on a priori $L_\infty$-bounds for the solutions and its gradient do not apply so that a new approach is needed. Various necessary or sufficient conditions are obtained on the data for existence. Relationship between existence of supersolutions and solutions is considered. Finally, sharp uniqueness results for weak solutions are given.