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

Abstract

In this paper we continue the analysis of the blow-up of low energy sign-changing solutions of semilinear elliptic equations with critical Sobolev exponent, started in [M. Ben Ayed, K. El Mehdi, F. Pacella, Blow-up and nonexistence of sign-changing solutions to the Brezis-Nirenberg problem in dimension three, Ann. Inst. H. Poincaré Anal. Non Linéaire, in press]. In addition we prove axial symmetry results for the same kind of solutions in a ball.

Classification of low energy sign-changing solutions of an almost critical problem

“…According to Theorem 1.5, we know that, for ε close to 0, both points a ε,i are bounded away from the origin and they lie on different sides with respect to T , where T is any hyperplane passing through the origin but not containing a ε,1 . Arguing now as in the proof of Lemma 4.1 of [8], we see that the points a ε,i lie on the same line passing through the origin. Lastly, the proof of the other statements of Theorem 1.7 is exactly the same as that of Theorem 1.5 of [8], so we omit it.…”

“…Arguing as in the proof of Theorem 1.1 of [8], we obtain that there exist a ε,1 , a ε,2 , μ ε,1 and μ ε,2 such that, as ε → 0,…”

The Ljapunov–Schmidt Reduction for Some Critical Problems

“…Moreover, for low dimensions n = 4, 5, 6, in [23] they proved the existence of at least n + 1 pairs of solutions provided ǫ is small enough. Ben Ayed-El Mehdi-Pacella in [7,8] studied the blow up of the low energy sign-changing solutions of problems (AC)ǫ and (BN )ǫ as ǫ goes to zero and they classified these solutions according to the concentration speeds of the positive and negative part. In [12] Castro-Clapp proved the existence of one pair of solutions in a symmetric domain which change sign exactly once, provided n ≥ 4 and ǫ is small enough.…”

“…Multi-bubbles for the Brezis-Nirenberg problem (BN )ǫ and the "slightly sub-critical" problem (AC)ǫ when ǫ is positive. According to Theorem 3.6, solutions to these problems are generated by C 1 −stable critical points of the function Ψ defined in (8) and (9) when ǫ > 0.…”

“…According to Theorem 3.6, solutions to these problems are generated by C 1 −stable critical points of the function Ψ defined in (8) and (9) when ǫ < 0.…”