Nontrivial solutions for p-Laplacian systems

• DOI: 10.1016/j.jmaa.2006.07.072
• Publication Date: 2006-08-25T21:40:24Z
• Authors: D.D. Hai; Haiyan Wang
• Keywords: Schauder Fixed-Point Theorem; Elliptic system; p-Laplacian; 0101 mathematics; 01 natural sciences

AbstractThe 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>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‖=∑i=1n|ui|, we prove that the problem has a nontrivial nonnegative solution for small λ>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>0. A nonexistence result is also obtained.