A remark on the existence of viscosity solutions for quasilinear elliptic equations

In the present paper we study the existence of viscosity solutions of the Dirichlet problem for quasilinear elliptic equations of the form-∑i,j=1naij(x)uxixj+b(x,u,Du)=0,where a(x), b(x,u,p) are continuous functions and the function b(x,u,p) has an arbitrary growth with respect to p. Under some structure restrictions on b(x,u,p) suitable subsolution and supersolution satisfying boundary conditions are constructed. The existence and uniqueness of viscosity solutions is obtained by Perron's method under assumption that the strong comparison result holds.