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

AbstractConsider the equation−Δpu=λg(x)|u|p−2u+f(x)|u|p*−2u ((1))in RN, where 1<p<Nandp*=Np/(N−p) is the critical Sobolev exponent. Letλ+1>0 be the principal eigenvalue of−Δpu=λg(x)|u|p−2uinRN,∫RNg(x)|u|p>0, ((2))withu+1>0 the associated eigenfunction. We prove: (i) Equation (1) has at least one positive solution ifλ∈(0,λ+1). (ii) Suppose ∫RNf(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)