SOME RESULTS ABOUT THE EXISTENCE OF A 2ND POSITIVE SOLUTION IN A QUASI-LINEAR CRITICAL PROBLEM

-Delta(p)u = -div(\del u\(p-2)del u) = lambda\u\(q-2)u+\u\(p*-2)u in Omega, u\delta Omega = 0, where Omega subset of R(N) is a smooth bounded domain, 1 < q < p < N, lambda > 0, p* = Np/(N-p). In this work we prove the existence of lambda(0) such that for 0 < lambda < lambda(0), the problem (P-lambda) has at least two positive solutions if either 2N/(N + 2) < p < 3 and 1 < q < p, or p greater than or equal to 3 and p > q > p* - 2/(p - 1).