Neumann problems of semilinear elliptic equations involving critical Sobolev exponents

AbstractIn this paper we study the existence of positive solutions to the equation Δu + up + f(x, u) = 0 under the Neumann boundary condition Dggu + α(x)u = 0, where p = (n + 2)(n − 2), f(x, u) is a lower order perturbation of up at infinity. When α(x) = 0, we prove the existence of a positive solution provided limu → 0 f(x, u)u = a(x) ⩽ 0, a(x) − 0, and f(x, u) ⩾ −Au − Buq for some constants A, B ⩾ 0, q ϵ (1, n(n − 2)). For general α(x), we prove the existence under an additional assumption on the boundary ∂Ω