Some Results about the Existence of a Second Positive Solution in a Quasilinear Critical Problem

Consider the problem $(P_\lambda ){-\Delta _pu\equiv -div(|\triangledown u|^{p-2} \triangledownu)=\lambda |u|^{q-2}u+|u|^{p^{*-2}}u$ in Ω, $u|_{\delta \Omega }=0$ , where Ω ⊂ RN is a smooth bounded domain, 1 < q < p < N, λ > 0, p* = Np/(N − p). In this work we prove the existence of λ0 such that for 0 < λ < λ0, the problem (Pλ) has at least two positive solutions if either 2N/(N + 2) < p < 3 and 1 < q < p, or p ≥ 3 and p > q > p* − 2/(p − 1).