Quasilinear elliptic equations with VMO coefficients

Abstract:

Strong solvability and uniqueness in Sobolev space ${W^{2,n}}(\Omega )$ are proved for the Dirichlet problem \[ \left \{ {_{u = \varphi \quad {\text {on}}\partial \Omega .}^{{a^{ij}}(x,u){D_{ij}}u + b(x,u,Du) = 0\quad {\text {a}}{\text {.e}}{\text {.}}\Omega }} \right .\] It is assumed that the coefficients of the quasilinear elliptic operator satisfy Carathéodory’s condition, the ${a^{ij}}$ are $V M O$ functions with respect to $x$, and structure conditions on $b$ are required. The main results are derived by means of the Aleksandrov-Pucci maximum principle and Leray-Schauder’s fixed point theorem via a priori estimate for the ${L^{2n}}$-norm of the gradient.