On certain nonlinear elliptic systems with indefinite terms.

Abstract

The purpose of this article is to nd positive solutions to the system ð€€€ pu = m(x)@H @ u(u; v) in

ð€€€ qv = m(x)@H @ v(u; v) in (1:1) u = v = 0 on@

where is a bounded regular domain of RN; with a smooth boundary @ ; pu := div (j ru jpð€€€2 ru) is the pð€€€ Laplacian with 1 < p < N;m is a continuous function on which changes sign , and H is a potential function which will be speci ed later . The case where the sign of m does not change has been studied by F . de Th e lin and J . V e lin [ 9 ] . These authors treat the system ( 1 . 1 ) with a function H having the following properties HThere(x; uexists ; v) CC(j>0 ujp0 ; + for j v jall0q ) x 2 ; for all (u; v) 2 D3 such that 0 There exists C0 > 0; for all x 2 ; for all (u; v) 2 D2 such that H(x; u; v) C0 There exists a positive function a in L1( ); such that for each x 2 and (u; v) 2 D1 \ R2 +;H(x; u; v) = a(x)u +1v +1;