In this paper, we consider the following Dirichlet problem for the fully nonlinear elliptic equation of Grad-Mercier type under asymptotic convexity conditions \begin{ equation*} \left\{ \begin{ array}{ ll} F(D^2u(x),Du(x),u(x),x)=g(|\{ y\in \Omega:u(y)\ge u(x)\}|)+f(x) & \text{ in } \Omega, u=\psi &\text{ on } \partial \Omega. \end{ array} \right. \end{ equation*} In order to overcome the non-convexity of the operator $F$ and the nonlocality of the nonhomogeneous term $g$, we apply the compactness methods and frozen technique to prove the existence of the $W^{ 2,p}$-viscosity solutions and the global $W^{ 2,p}$ estimate. As an application, we derive a Cordes-Nirenberg type continuous estimate up to boundary. Furthermore, we establish a global BMO estimate for the second derivatives of solutions by using an asymptotic approach, thereby refining the borderline case of Calder\'{ o}n-Zygmund estimates.
Categories: math.AP