Nontrivial solutions for p-Laplacian systems

The paper deals with the existence and nonexistence of nontrivial nonnegative solutions for the sublinear quasilinear system div (vertical bar del u(i)vertical bar(p-2)del u(i)) +lambda f(i) (u(1),..., u(n)) = 0 in Omega, u(i) = 0 on partial derivative Omega, i = 1,...,n, where p > 1, Omega is a bounded domain in R-N (N >= 2) with smooth boundary, and f(i), i = 1,..., n, are continuous, nonnegative functions. Let u = (u(1),..., u(n)), parallel to u parallel to = Sigma(n)(i=1) vertical bar u(i)vertical bar, we prove that the problem has a nontrivial nonnegative solution for small lambda > 0 if one of lim(parallel to u parallel to -> 0) fi(u)/parallel to u parallel to(p-1) is infinity. If, in addition, all lim(parallel to u parallel to ->infinity) fi(u)/parallel to u parallel to(p-1) is zero, we show that the problem has a nontrivial nonnegative solution for all lambda > 0. A nonexistence result is also obtained. (c) 2006 Elsevier Inc. All rights reserved.