An application of generalized Morrey spaces to unique continuation property of the quasilinear elliptic equations

Abstract

In this paper, we study nonnegative weak solutions of the quasilinear elliptic equation $ \text{div}(A(x, u, \nabla u)) = B(x, u, \nabla u) $, in a bounded open set $ \Omega $, whose coefficients belong to a generalized Morrey space. We show that $ \log (u + \delta) $, for $ u $ a nonnegative solution and $ \delta $ an arbitrary positive real number, belongs to $ \text{BMO}(B) $, where $ B $ is an open ball contained in $ \Omega $. As a consequence, this equation has the strong unique continuation property. For the main proof, we use approximation by smooth functions to the weak solutions to handle the weak gradient of the composite function which involves the weak solutions and then apply Fefferman's inequality in generalized Morrey spaces, recently proved by Tumalun et al. [1].

Keywords

• quasilinear elliptic equations
• generalized morrey spaces
• fefferman's inequality
• bounded mean oscillation
• strong unique continuation property