Nontrivial solutions for p-Laplacian systems

The paper deals with the existence and nonexistence of nontrivial nonnegative solutions for the sublinear quasilinear system div |∇ui |p−2∇ui +λfi(u1, . . . , un) =0 inΩ, ui =0 on∂Ω, i = 1, . . . , n, where p > 1, Ω is a bounded domain in RN (N 2) with smooth boundary, and fi , i = 1, . . . , n, are continuous, nonnegative functions. Let u = (u1, . . . , un), u = n i=1 |ui |, we prove that the problem has a nontrivial nonnegative solution for small λ > 0 if one of lim u →0 fi (u) u p−1 is infinity. If, in addition, all lim u →∞ fi (u) u p−1 is zero, we show that the problem has a nontrivial nonnegative solution for all λ > 0. A nonexistence result is also obtained. © 2006 Elsevier Inc. All rights reserved.