We are concerned with the existence and the asymptotic analysis when the parameter εtends to 0of solutions with multiple concentration for the following almost critical problem: −Δu = u^{(N+2)/(N−2)}+ε in Ω, u>0 inΩ, u= 0 on ∂Ω, where Ωis a bounded domain in RNwith a smooth boundary and N≥3. Weare interested in concentration phenomena from the supercritical side ε →0+. In particular we prove that, if Ωhas a small and not necessarily symmetric hole, then for any fixed odd integer k≥3 there exists a family of solutions which develops a multiple bubble-shape as ε →0+, blowing up at kdifferent points in Ω. This extends the previous result by Del Pino, Felmer and Musso [13], where solutions with a two-bubbleprofile are constructed.

D'Aprile, T. (2016). Multi-bubble solutions for a slightly supercritical elliptic problem in a domain with a small hole. JOURNAL DE MATHÉMATIQUES PURES ET APPLIQUÉES, 105, 558-602 [10.1016/j.matpur.2015.11.008].

Multi-bubble solutions for a slightly supercritical elliptic problem in a domain with a small hole

D'APRILE, TERESA CARMEN
2016

Abstract

We are concerned with the existence and the asymptotic analysis when the parameter εtends to 0of solutions with multiple concentration for the following almost critical problem: −Δu = u^{(N+2)/(N−2)}+ε in Ω, u>0 inΩ, u= 0 on ∂Ω, where Ωis a bounded domain in RNwith a smooth boundary and N≥3. Weare interested in concentration phenomena from the supercritical side ε →0+. In particular we prove that, if Ωhas a small and not necessarily symmetric hole, then for any fixed odd integer k≥3 there exists a family of solutions which develops a multiple bubble-shape as ε →0+, blowing up at kdifferent points in Ω. This extends the previous result by Del Pino, Felmer and Musso [13], where solutions with a two-bubbleprofile are constructed.
Pubblicato
Rilevanza internazionale
Articolo
Esperti anonimi
Settore MAT/05 - Analisi Matematica
English
Con Impact Factor ISI
Slightly supercritical exponent; Multi-bubble solutions; Finite-dimensional reduction; Min-max argument
http://www.sciencedirect.com/science/article/pii/S0021782415001580
D'Aprile, T. (2016). Multi-bubble solutions for a slightly supercritical elliptic problem in a domain with a small hole. JOURNAL DE MATHÉMATIQUES PURES ET APPLIQUÉES, 105, 558-602 [10.1016/j.matpur.2015.11.008].
D'Aprile, Tc
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/2108/184277
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