Multiplicity of Positive Solutions for Some Quasilinear Elliptic Equation in RNwith Critical Sobolev Exponent (original) (raw)

Consider the equation &2 p u=*g(x) |u| p&2 u+f(x) |u| p*&2 u (1) in R N , where 1<p<N and p*=NpÂ(N&p) is the critical Sobolev exponent. Let * + 1 >0 be the principal eigenvalue of &2 p u=*g(x) |u| p&2 u in R N , | R N g(x) |u| p >0, (2) with u + 1 >0 the associated eigenfunction. We prove: (i) Equation (1) has at least one positive solution if * # (0, * + 1). (ii) Suppose R N f (x)(u + 1) p* <0. Then there exists * 0 >* + 1 such that (1) has at least one positive solution for * # [* + 1 , * 0). Moreover, if p 2, there exists * 0 >* + 1 such that (1) has at least two positive solutions for * # (* + 1 , * 0).