In this paper we show that there exists a family of domains Ω_e ⊂ R^N with N≥ 2, such that the stable solution of the problem -∆u= g(u) in Ω_e u>0 in Ω_e u=0 on ∂Ω_e admits k critical points with k≥ 2. Moreover the sets Ω_e are star-shaped and ``close'' to a strip as e→0. Next, if g(u)=1 and N≥ 3 we exhibit a family of domains Ω_e with positive mean curvature and solutions u_e which have k critical points with k≥ 2. In this case, the domains Ω_e turn out to be ``close'' to a cylinder as e→0.

On the number of critical points of stable solutions in bounded strip-like domains / De Regibus, Fabio; Grossi, Massimo. - In: JOURNAL OF DIFFERENTIAL EQUATIONS. - ISSN 0022-0396. - 306:(2022), pp. 1-27. [10.1016/j.jde.2021.10.028]

On the number of critical points of stable solutions in bounded strip-like domains

De Regibus, Fabio;Grossi, Massimo
2022

Abstract

In this paper we show that there exists a family of domains Ω_e ⊂ R^N with N≥ 2, such that the stable solution of the problem -∆u= g(u) in Ω_e u>0 in Ω_e u=0 on ∂Ω_e admits k critical points with k≥ 2. Moreover the sets Ω_e are star-shaped and ``close'' to a strip as e→0. Next, if g(u)=1 and N≥ 3 we exhibit a family of domains Ω_e with positive mean curvature and solutions u_e which have k critical points with k≥ 2. In this case, the domains Ω_e turn out to be ``close'' to a cylinder as e→0.
2022
critical points; convexity
01 Pubblicazione su rivista::01a Articolo in rivista
On the number of critical points of stable solutions in bounded strip-like domains / De Regibus, Fabio; Grossi, Massimo. - In: JOURNAL OF DIFFERENTIAL EQUATIONS. - ISSN 0022-0396. - 306:(2022), pp. 1-27. [10.1016/j.jde.2021.10.028]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11573/1619913
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