A strong comparison principle for positive solutions of degenerate elliptic equations

Abstract

A strong comparison principle (SCP, for brevity) is obtained for nonnega-tive weak solutions u 2 W 1,p 0 (⌦) of the following class of quasilinear elliptic boundary value problems, (P) div(a(x, ru)) b(x, u) = f (x) in ⌦; u = 0 on @⌦. Here, p 2 (1, 1) is a given number, ⌦ is a bounded domain in IR N with a connected C 2-boundary, a(x, ru) and b(x, u) are slightly more general than the functions a 0 (x)|ru| p2 ru and b 0 (x)|u| p2 u, respectively, with a 0 const > 0 and b 0 0 in L 1 (⌦), and 0  f 2 L 1 (⌦). Validity of the SCP is investigated also in the case when b 0  0 depending upon whether p  2 or p > 2. The methods of proofs are new.