Sign changing bump solutions for Schrödinger equations involving critical growth and indefinite potential wells

In this paper, we consider the following Schrödinger equations with critical growth−Δu+(λa(x)−δ)u=|u|2⁎−2u,x∈RN, where N≥4, 2⁎ is the critical Sobolev exponent, a(x)≥0 and its zero sets are not empty, λ>0 is a parameter, δ>0 is a constant such that the operator (−Δ+λa(x)−δ) might be indefinite for λ large. We prove that if the zero sets of a(x) have several isolated connected components Ω1,⋯,Ωk such that the interior of Ωi(i=1,2,…,k) is not empty and ∂Ωi(i=1,2,…,k) is smooth. Then for λ sufficiently large, the equation admits, for any i∈{1,2,⋯,k}, a solution which is trapped in a neighborhood of Ωi. The key ingredients of the paper are using a flow argument and a combination of global linking and local linking.