Blow-up and nonexistence of sign changing solutions to the Brezis–Nirenberg problem in dimension three

Abstract

In this paper we study low energy sign changing solutions of the critical exponent problem (P λ): − u = u 5 + λu in Ω, u = 0 on ∂Ω, where Ω is a smooth bounded domain in R 3 and λ is a real positive parameter. We make a precise blow-up analysis of this kind of solutions and prove some comparison results among some limit values of the parameter λ which are related to the existence of positive or of sign changing solutions.

Blow-up and symmetry of sign-changing solutions to some critical elliptic equations

“…Obviously the solutions considered in [6] in dimension 3 and the ones satisfying the hypotheses of Theorem 1.2 for n 4 verify assumptions (1)-(5). Theorem 1.6.…”

“…Since h λ is bounded and d (n−2)/2 λ,1 u λ (a λ,1 ) → ∞, we derive that 0 is an isolated blow-up point of (w λ ). Now we can proceed as in [6,Appendix] to prove that 0 is an isolated simple blow-up point of (w λ ). In fact, the proof of this assertion is almost the same as in [6,Proposition 5.9].…”

“…The method used here is different from that of [6] and while some initial results hold for any dimension n 3 without any extra assumptions, a more precise blow-up analysis requires the a priori assumption that the concentration speeds of the two concentration points are comparable, while in [6] this result was derived without any extra hypothesis.…”

“…Though the case n = 3 is usually considered more difficult and results true in dimension 3 are expected to hold also in higher dimensions with similar or even simpler proofs, the proof of [6] only works in dimension 3, because in applying Pohozaev's identity and getting the convergence of certain integrals, the role of the dimension is crucial.…”

“…In [6] we investigated the asymptotic behavior of low energy sign-changing solutions of (1) in dimension 3, as the parameter λ goes to the limit parameter λ(Ω) = inf λ ∈ R: (1) has a sign-changing solution u λ with u λ 2 λ…”