Multiplicity of Positive Solutions for Some Quasilinear Elliptic Equation in Nwith Critical Sobolev Exponent

Consider the equation−Δpu=λg(x) up−2 u+f(x) up*−2 u ((1))in N, where 10 be the principal eigenvalue of withu+1>0 the associated eigenfunction. We prove: (i) Equation (1) has at least one positive solution ifλ (0, λ+1). (ii) Suppose ∫ N f(x)(u+1)p*<0. Then there existsλ0>λ+1such that (1) has at least one positive solution forλ [λ+1, λ0). Moreover, ifp 2, there exists λ0>λ+1such that (1) has at least two positive solutions forλ (λ+1, λ0).