MULTIPLICITY OF POSITIVE SOLUTIONS FOR A CLASS OF CONCAVE-CONVEX ELLIPTIC EQUATIONS WITH CRITICAL GROWTH

In this article, the following concave and convex nonlinearities elliptic equations involving critical growth is considered,{-△u=g（x）|u|2*-2u+λf（x）|u|q-2u,x∈Ω,u=0,x∈■Ω where Ω■RN（N≥3） is an open bounded domain with smooth boundary, 1 < q < 2, λ > 0.2*=2 N/（N-2）is the critical Sobolev exponent,f∈L2*/（2*-q）(Ω)is nonzero and nonnegative,and g ∈ C（■） is a positive function with k local maximum points. By the Nehari method and variational method,k+1 positive solutions are obtained. Our results complement and optimize the previous work by Lin [MR2870946, Nonlinear Anal. 75（2012） 2660-2671].