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In this paper, we study nonnegative weak solutions of the quasilinear elliptic equation div(A(x,u,∇u))=B(x,u, ∇u), in a bounded open set Ω, whose coefficients belong to a generalized Morrey space. We show that log(u + δ), for u a nonnegative solution and δ an arbitrary positive real number, belongs to BMO(B), where B is an open ball contained in Ω. 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]. [ABSTRACT FROM AUTHOR]

In this paper we prove Fefferman's inequalities associated to potentials belonging to a generalized Morrey space or a Stummel class. We also show that the logarithm of a non-negative weak solution to a second order elliptic partial differential equation with potential in a generalized Morrey space or a Stummel class, under some assumptions, belongs to the bounded mean oscillation class. As a consequence, this elliptic partial differential equation has the strong unique continuation property. An example of an elliptic partial differential equation with potential in a Morrey space or a Stummel class which does not satisfy the strong unique continuation is presented. [ABSTRACT FROM AUTHOR]

In this paper, we show that the Morrey spaces Mpq1 (Rn) cannot be contained in the weak Morrey spaces wMpq2 (Rn) for q1 ≠ q2. We also show that the vanishing Morrey spaces VMpq (Rn) are not empty and properly contained in the Morrey spaces Mpq (Rn). [ABSTRACT FROM AUTHOR]

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