Nontrivial solutions for p-Laplacian systems
نویسندگان
چکیده
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 in Ω, 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.
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