Existence of solutions of quasilinear elliptic equations in \(R^ n\) (Q1820310)

The author studies in \(R^ n\) a quasilinear elliptic equation \[ (1)\quad Lu\equiv -\sum_{1\leq i,j\leq n}a_{ij}(x)D_ iD_ ju=f(x,u,\nabla u). \] Let functions v,w exist such that Lv\(\geq f(x,v,\nabla v)\), Lw\(\leq f(x,w,\nabla w)\) and w(x)\(\leq v(x)\). Under certain conditions on f(x,u,p) the author proves the existence of a solution u of (1) such that w(x)\(\leq u(x)\leq v(x)\) \((x\in R^ n)\).