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

Let Ω be a smooth bounded domain in , with N 5, a>0, α 0 and . We show that the exponent plays a critical role regarding the existence of least energy (or ground state) solutions of the Neumann problem Namely, we prove that when there exists an α0>0 such that the problem has a least energy solution if α<α0 and has no least energy solution if α>α0.