A weak comparison principle for some quasilinear elliptic operators: it compares functions belonging to different spaces

summary:We shall prove a weak comparison principle for quasilinear elliptic operators $-{\rm div}(a(x,\nabla u))$ that includes the negative $p$-Laplace operator, where $a\colon \Omega \times \Bbb R^N \rightarrow \Bbb R^N$ satisfies certain conditions frequently seen in the research of quasilinear elliptic operators. In our result, it is characteristic that functions which are compared belong to different spaces