Perturbation of Δu+u(N+2)/(N−2)=0, the Scalar Curvature Problem in RN, and Related Topics

Abstract

Some nonlinear elliptic equations on R N which arise perturbing the problem with the critical Sobolev exponent are studied. In particular, some results dealing with the scalar curvature problem in R N are given.1999 Academic Press

On the Yamabe problem and the scalar curvature problems under boundary conditions

“…n−2 dσ and can be faced by means of a perturbation method in critial point theory discussed in [1]. Finally, in the Appendix we prove some technical Lemmas.…”

“…Arguments of this kind has been emploied, e.g. in [1]. The first step is to construct, for g ∈ G ε , a perturbed manifold Z c g Z c which is a natural constraint…”

“…In general, the preceding arguments give rise to the following result, see [1]. Let I ε (u) = I 0 (u) + O(ε) denote a C 2 functional and suppose that I 0 has an n-dimensional manifold Z of critical points satisfying (i) − (ii) of Lemma 3.…”

“…More recently, the existence of a solution of (1) has been proved in [14,15] under the assumption that (B, g) is of positive type (for a definition see [14]) and satisfies one of the following assumptions:…”

Positive or sign-changing solutions for a critical semilinear nonlocal equation

“…Before our study on this problem, we would like draw the reader's attention to some recent results on the multiplicity of positive solutions to equation (1.3). Amrosetti, Azorero and Peral [1], and Cao, Noussair and Yan [7] proved the existence of two or many positive solutions if K is a perturbation of the constant, i.e. K = K 0 + εh(x), 0 < ε ≪ 1.…”