Uniqueness results for nonlinear elliptic equations with a lower order term

We prove the uniqueness of the weak solution of noncoercive nonlinear elliptic problems whose model is {u is an element of W-0(1,p) (ohm), -div(a(x)(1 + vertical bar del u vertical bar(2))((p-2)/2)del u) + b(x) (1 + vertical bar del u vertical bar(2))((sigma + 1)/2) = f in D ' (ohm).} where 0 is a bounded open subset of R-N, N >= 2,p satisfies p >= 2N/(N + 1), a is afunction belonging to L infinity(ohm)sucht that a(x) >= alpha > 0, f belongs to the dual space W-1,(p)'(ohm), b belongs to some Lebesgue space L-r(ohm) with r >= r*( N, p) and alpha-belongs to the interval [0,sigma*(N,p,r)], with sigma*(N,p,r) and r*(N, p,) functions which are specified below. (c) 2005 Elsevier Ltd. All rights reserved.