Yessine DAMMAK

A nonexistence result for blowing up sign-changing solutions of the Brezis–Nirenberg-type problem

We consider the Brezis–Nirenberg problem: $−△u = |u|p−1u ± εu in Ω, with u = 0 on ∂Ω$, where Ω is asmooth bounded domain in Rn, n ≥ 4, p + 1 = 2n/(n − 2) is the critical Sobolev exponent, and ε > 0 is a positiveparameter. The main result of this paper shows that if n ≥ 4 there are no sign-changing solutions uε of (P−ε) with twopositive and one negative blow up points.

PDF

___

Atkinson FV, Brezis H, Peletier LA. Nodal solutions of elliptic equations with critical Sobolev exponents. J Differ Equations 1990; 85: 151-170.
Atkinson FV, Brezis H, Peletier LA. Solutions d’équations elliptiques avec exposant de Sobolev critique qui changent de signe. C R Math Paris Sér I Math 1988; 306: 711-714.
Bahri A. Critical points at infinity in some variational problems. Pitman Res Notes Math Ser, Vol 182. Harlow, UK: Longman, 1989.
Bahri A, Xu Y. Recent Progress in Conformal Geometry. London, UK: Imperial College Press, 2007
Ben Ayed M, Chtioui H, Hammami M. A Morse lemma at infinity for Yamabe type problems on domains. Ann I H Poincare-AN 2003; 20: 543-577.
Ben Ayed M, El Mehdi K, Pacella F. Blow-up and symmetry of sign-changing solutions to some critical elliptic equations. J Differ Equations 2006; 230: 771-795.
Ben Ayed M, El Mehdi K, Pacella F. Blow-up and nonexistence of sign-changing solutions to the Brezis-Nirenberg problem in dimension three. Ann I H Poincare-AN 2006; 23: 567-589.
Brezis H, Nirenberg L. Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Commun Pur Appl Math 1983; 36: 437-477.
Capozzi A, Fortunato D, Palmieri G. An existence result for nonlinear elliptic problems involving critical Sobolev exponent. Ann I H Poincare-AN 1985; 2: 463-470.
Clapp M, Weth T. Multiple solutions for the Brezis-Nirenberg problem. Adv Differential Equ 2005; 10: 463-480.
Dammak Y. A nonexistence result for low energy sign-changing solutions of the Brezis-Nirenberg problem in dimensions 4, 5 and 6. J Differ Equations 2017; 263: 7559-7600.
Hammami M, Ismail H. Existence and non-existence of sign-changing solutions to elliptic critical equations. Turk J Math 2016; 40: 517-539.
acopetti A, Pacella F. A nonexistence result for sign-changing solutions of the Brezis-Nirenberg problem in low dimensions. J Differ Equations 2015; 258: 4180-4208.
Iacopetti A, Vaira G. Sign-changing tower of bubbles for the Brezis-Nirenberg problem. Commun Contemp Math 2016; 18: no. 01.
Musso M, Pistoia A. Double blow up for a Brezis-Niremberg type problem. Commun Contemps Math 2003; 5: 775-802.
Rey O. The role of the Green’s function in a non-linear elliptic equation involving the critical Sobolev exponent. J Funct Anal 1990; 89: 1-52.
Schechter M, Zou W. On the Brézis-Niremberg problem. Arch Ration Mech An 2010: 337-356.