Multiplicity of positive solutions for a class of concave-convex elliptic equations with critical growth

Abstract

In this article, the following concave and convex nonlinearities elliptic equations involving critical growth is considered, {-Δu=g(x)|u|2*-2u+λf(x)|u|q-2u,x∈Ωu=0,x∈∂Ω,where Ω ⊂ RN (N ≥ 3) is an open bounded domain with smooth boundary, 1 < q < 2,λ > 0. 2*=2NN-2 is the critical Sobolev exponent, f∈L2*2*-q(Ω) is nonzero and nonnegative, and g ∈ C (Ω¯) 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].