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

摘要

Let Omega be a smooth bounded domain in R-N, with Ngreater than or equal to5, a > 0, alphagreater than or equal to0 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 -Deltau + au = u(2*-1) - alphau(q-1) in Omega, u > 0 in Omega, partial derivativeu/partial derivativev = 0 on partial derivativeOmega. Namely, we prove that when q = 2(N-1)/N-2 there exists an alpha(0) > 0 such that the problem has a least energy solution if alpha alpha(0). (C) 2002 Elsevier Science (USA). All rights reserved. [References: 21]