Existence and nonexistence of least energy solutions of the Neumann problem for a semilinear elliptic equation with critical Sobolev exponent and a critical lower-order perturbation

Abstract

Let Ω be a smooth bounded domain in R N , with N⩾5, a>0, α⩾0 and 2 ∗= 2N N−2 . We show that the exponent q= 2(N−1) N−2 plays a critical role regarding the existence of least energy (or ground state) solutions of the Neumann problem − Δu+au=u 2 ∗−1 −αu q−1 in Ω, u>0 in Ω, ∂u ∂ν =0 on ∂Ω. Namely, we prove that when q= 2(N−1) N−2 there exists an α 0>0 such that the problem has a least energy solution if α< α 0 and has no least energy solution if α> α 0.