Nontrivial solutions for p-Laplacian systems

The paper deals with the existence and nonexistence of nontrivial nonnegative solutions for the sublinear quasilinear systemdiv(|∇ui|p−2∇ui)+λfi(u1,…,un)=0in Ω,ui=0on ∂Ω,i=1,…,n, where p>1p>1, Ω   is a bounded domain in RNRN (N⩾2N⩾2) with smooth boundary, and fifi, i=1,…,ni=1,…,n, are continuous, nonnegative functions. Let u=(u1,…,un)u=(u1,…,un), ‖u‖=∑i=1n|ui|, we prove that the problem has a nontrivial nonnegative solution for small λ>0λ>0 if one of lim‖u‖→0fi(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>0λ>0. A nonexistence result is also obtained.