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

In this paper, we study nonnegative weak solutions of the quasilinear elliptic equation div(A( x, u, del u)) = B( x, u, del 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 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].