A blowing-up branch of solutions for a mean field equation
Abstract We consider the equation $$ -\Delta u= \lambda\bigg(\frac{e^u}{\int_{\Omega} e^u}- \frac{1}{|\Omega|}\bigg), \quad u \in H^1_0 (\Omega).$$If Ω is of class C2, we show that this problem has a non-trivial solution uλ for each λ ∊ (8 π, $ λ^{*} $). The value $ λ^{*} $ depends on the domain and...
Ausführliche Beschreibung
Gespeichert in:
| Autor*in: |
Lucia, Marcello [verfasserIn] |
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| Format: |
Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2006 |
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| Schlagwörter: |
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| Anmerkung: |
© Springer Science + Business Media, Inc. 2006 |
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| Übergeordnetes Werk: |
Enthalten in: Calculus of variations and partial differential equations - Springer-Verlag, 1993, 26(2006), 3 vom: 15. März, Seite 313-330 |
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| Übergeordnetes Werk: |
volume:26 ; year:2006 ; number:3 ; day:15 ; month:03 ; pages:313-330 |
| Links: |
|---|
| DOI / URN: |
10.1007/s00526-006-0007-3 |
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OLC207519131X |
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| 520 | |a Abstract We consider the equation $$ -\Delta u= \lambda\bigg(\frac{e^u}{\int_{\Omega} e^u}- \frac{1}{|\Omega|}\bigg), \quad u \in H^1_0 (\Omega).$$If Ω is of class C2, we show that this problem has a non-trivial solution uλ for each λ ∊ (8 π, $ λ^{*} $). The value $ λ^{*} $ depends on the domain and is bounded from below by 2 j02 π, where j0 is the first zero of the Bessel function of the first kind of order zero ($ λ^{*} $≥ 2 j02 π > 8 π). Moreover, the family of solution uλ blows-up as λ → 8 π. | ||
| 650 | 4 | |a Mean field equations | |
| 650 | 4 | |a Moser-Trudinger inequality | |
| 650 | 4 | |a Mountain pass theorem | |
| 650 | 4 | |a Faber-Krahn inequality | |
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10.1007/s00526-006-0007-3 doi (DE-627)OLC207519131X (DE-He213)s00526-006-0007-3-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Lucia, Marcello verfasserin aut A blowing-up branch of solutions for a mean field equation 2006 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Science + Business Media, Inc. 2006 Abstract We consider the equation $$ -\Delta u= \lambda\bigg(\frac{e^u}{\int_{\Omega} e^u}- \frac{1}{|\Omega|}\bigg), \quad u \in H^1_0 (\Omega).$$If Ω is of class C2, we show that this problem has a non-trivial solution uλ for each λ ∊ (8 π, $ λ^{*} $). The value $ λ^{*} $ depends on the domain and is bounded from below by 2 j02 π, where j0 is the first zero of the Bessel function of the first kind of order zero ($ λ^{*} $≥ 2 j02 π > 8 π). Moreover, the family of solution uλ blows-up as λ → 8 π. Mean field equations Moser-Trudinger inequality Mountain pass theorem Faber-Krahn inequality Enthalten in Calculus of variations and partial differential equations Springer-Verlag, 1993 26(2006), 3 vom: 15. März, Seite 313-330 (DE-627)165669977 (DE-600)1144181-1 (DE-576)033045690 0944-2669 nnns volume:26 year:2006 number:3 day:15 month:03 pages:313-330 https://doi.org/10.1007/s00526-006-0007-3 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_40 GBV_ILN_70 GBV_ILN_100 GBV_ILN_2002 GBV_ILN_2004 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2018 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2088 GBV_ILN_2409 GBV_ILN_4027 GBV_ILN_4036 GBV_ILN_4277 GBV_ILN_4305 GBV_ILN_4307 GBV_ILN_4313 AR 26 2006 3 15 03 313-330 |
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10.1007/s00526-006-0007-3 doi (DE-627)OLC207519131X (DE-He213)s00526-006-0007-3-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Lucia, Marcello verfasserin aut A blowing-up branch of solutions for a mean field equation 2006 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Science + Business Media, Inc. 2006 Abstract We consider the equation $$ -\Delta u= \lambda\bigg(\frac{e^u}{\int_{\Omega} e^u}- \frac{1}{|\Omega|}\bigg), \quad u \in H^1_0 (\Omega).$$If Ω is of class C2, we show that this problem has a non-trivial solution uλ for each λ ∊ (8 π, $ λ^{*} $). The value $ λ^{*} $ depends on the domain and is bounded from below by 2 j02 π, where j0 is the first zero of the Bessel function of the first kind of order zero ($ λ^{*} $≥ 2 j02 π > 8 π). Moreover, the family of solution uλ blows-up as λ → 8 π. Mean field equations Moser-Trudinger inequality Mountain pass theorem Faber-Krahn inequality Enthalten in Calculus of variations and partial differential equations Springer-Verlag, 1993 26(2006), 3 vom: 15. März, Seite 313-330 (DE-627)165669977 (DE-600)1144181-1 (DE-576)033045690 0944-2669 nnns volume:26 year:2006 number:3 day:15 month:03 pages:313-330 https://doi.org/10.1007/s00526-006-0007-3 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_40 GBV_ILN_70 GBV_ILN_100 GBV_ILN_2002 GBV_ILN_2004 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2018 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2088 GBV_ILN_2409 GBV_ILN_4027 GBV_ILN_4036 GBV_ILN_4277 GBV_ILN_4305 GBV_ILN_4307 GBV_ILN_4313 AR 26 2006 3 15 03 313-330 |
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10.1007/s00526-006-0007-3 doi (DE-627)OLC207519131X (DE-He213)s00526-006-0007-3-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Lucia, Marcello verfasserin aut A blowing-up branch of solutions for a mean field equation 2006 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Science + Business Media, Inc. 2006 Abstract We consider the equation $$ -\Delta u= \lambda\bigg(\frac{e^u}{\int_{\Omega} e^u}- \frac{1}{|\Omega|}\bigg), \quad u \in H^1_0 (\Omega).$$If Ω is of class C2, we show that this problem has a non-trivial solution uλ for each λ ∊ (8 π, $ λ^{*} $). The value $ λ^{*} $ depends on the domain and is bounded from below by 2 j02 π, where j0 is the first zero of the Bessel function of the first kind of order zero ($ λ^{*} $≥ 2 j02 π > 8 π). Moreover, the family of solution uλ blows-up as λ → 8 π. Mean field equations Moser-Trudinger inequality Mountain pass theorem Faber-Krahn inequality Enthalten in Calculus of variations and partial differential equations Springer-Verlag, 1993 26(2006), 3 vom: 15. März, Seite 313-330 (DE-627)165669977 (DE-600)1144181-1 (DE-576)033045690 0944-2669 nnns volume:26 year:2006 number:3 day:15 month:03 pages:313-330 https://doi.org/10.1007/s00526-006-0007-3 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_40 GBV_ILN_70 GBV_ILN_100 GBV_ILN_2002 GBV_ILN_2004 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2018 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2088 GBV_ILN_2409 GBV_ILN_4027 GBV_ILN_4036 GBV_ILN_4277 GBV_ILN_4305 GBV_ILN_4307 GBV_ILN_4313 AR 26 2006 3 15 03 313-330 |
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10.1007/s00526-006-0007-3 doi (DE-627)OLC207519131X (DE-He213)s00526-006-0007-3-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Lucia, Marcello verfasserin aut A blowing-up branch of solutions for a mean field equation 2006 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Science + Business Media, Inc. 2006 Abstract We consider the equation $$ -\Delta u= \lambda\bigg(\frac{e^u}{\int_{\Omega} e^u}- \frac{1}{|\Omega|}\bigg), \quad u \in H^1_0 (\Omega).$$If Ω is of class C2, we show that this problem has a non-trivial solution uλ for each λ ∊ (8 π, $ λ^{*} $). The value $ λ^{*} $ depends on the domain and is bounded from below by 2 j02 π, where j0 is the first zero of the Bessel function of the first kind of order zero ($ λ^{*} $≥ 2 j02 π > 8 π). Moreover, the family of solution uλ blows-up as λ → 8 π. Mean field equations Moser-Trudinger inequality Mountain pass theorem Faber-Krahn inequality Enthalten in Calculus of variations and partial differential equations Springer-Verlag, 1993 26(2006), 3 vom: 15. März, Seite 313-330 (DE-627)165669977 (DE-600)1144181-1 (DE-576)033045690 0944-2669 nnns volume:26 year:2006 number:3 day:15 month:03 pages:313-330 https://doi.org/10.1007/s00526-006-0007-3 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_40 GBV_ILN_70 GBV_ILN_100 GBV_ILN_2002 GBV_ILN_2004 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2018 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2088 GBV_ILN_2409 GBV_ILN_4027 GBV_ILN_4036 GBV_ILN_4277 GBV_ILN_4305 GBV_ILN_4307 GBV_ILN_4313 AR 26 2006 3 15 03 313-330 |
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10.1007/s00526-006-0007-3 doi (DE-627)OLC207519131X (DE-He213)s00526-006-0007-3-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Lucia, Marcello verfasserin aut A blowing-up branch of solutions for a mean field equation 2006 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Science + Business Media, Inc. 2006 Abstract We consider the equation $$ -\Delta u= \lambda\bigg(\frac{e^u}{\int_{\Omega} e^u}- \frac{1}{|\Omega|}\bigg), \quad u \in H^1_0 (\Omega).$$If Ω is of class C2, we show that this problem has a non-trivial solution uλ for each λ ∊ (8 π, $ λ^{*} $). The value $ λ^{*} $ depends on the domain and is bounded from below by 2 j02 π, where j0 is the first zero of the Bessel function of the first kind of order zero ($ λ^{*} $≥ 2 j02 π > 8 π). Moreover, the family of solution uλ blows-up as λ → 8 π. Mean field equations Moser-Trudinger inequality Mountain pass theorem Faber-Krahn inequality Enthalten in Calculus of variations and partial differential equations Springer-Verlag, 1993 26(2006), 3 vom: 15. März, Seite 313-330 (DE-627)165669977 (DE-600)1144181-1 (DE-576)033045690 0944-2669 nnns volume:26 year:2006 number:3 day:15 month:03 pages:313-330 https://doi.org/10.1007/s00526-006-0007-3 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_40 GBV_ILN_70 GBV_ILN_100 GBV_ILN_2002 GBV_ILN_2004 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2018 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2088 GBV_ILN_2409 GBV_ILN_4027 GBV_ILN_4036 GBV_ILN_4277 GBV_ILN_4305 GBV_ILN_4307 GBV_ILN_4313 AR 26 2006 3 15 03 313-330 |
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A blowing-up branch of solutions for a mean field equation |
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Lucia, Marcello |
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Calculus of variations and partial differential equations |
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Calculus of variations and partial differential equations |
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eng |
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500 - Science |
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2006 |
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Lucia, Marcello |
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Lucia, Marcello |
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10.1007/s00526-006-0007-3 |
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510 |
| title_sort |
a blowing-up branch of solutions for a mean field equation |
| title_auth |
A blowing-up branch of solutions for a mean field equation |
| abstract |
Abstract We consider the equation $$ -\Delta u= \lambda\bigg(\frac{e^u}{\int_{\Omega} e^u}- \frac{1}{|\Omega|}\bigg), \quad u \in H^1_0 (\Omega).$$If Ω is of class C2, we show that this problem has a non-trivial solution uλ for each λ ∊ (8 π, $ λ^{*} $). The value $ λ^{*} $ depends on the domain and is bounded from below by 2 j02 π, where j0 is the first zero of the Bessel function of the first kind of order zero ($ λ^{*} $≥ 2 j02 π > 8 π). Moreover, the family of solution uλ blows-up as λ → 8 π. © Springer Science + Business Media, Inc. 2006 |
| abstractGer |
Abstract We consider the equation $$ -\Delta u= \lambda\bigg(\frac{e^u}{\int_{\Omega} e^u}- \frac{1}{|\Omega|}\bigg), \quad u \in H^1_0 (\Omega).$$If Ω is of class C2, we show that this problem has a non-trivial solution uλ for each λ ∊ (8 π, $ λ^{*} $). The value $ λ^{*} $ depends on the domain and is bounded from below by 2 j02 π, where j0 is the first zero of the Bessel function of the first kind of order zero ($ λ^{*} $≥ 2 j02 π > 8 π). Moreover, the family of solution uλ blows-up as λ → 8 π. © Springer Science + Business Media, Inc. 2006 |
| abstract_unstemmed |
Abstract We consider the equation $$ -\Delta u= \lambda\bigg(\frac{e^u}{\int_{\Omega} e^u}- \frac{1}{|\Omega|}\bigg), \quad u \in H^1_0 (\Omega).$$If Ω is of class C2, we show that this problem has a non-trivial solution uλ for each λ ∊ (8 π, $ λ^{*} $). The value $ λ^{*} $ depends on the domain and is bounded from below by 2 j02 π, where j0 is the first zero of the Bessel function of the first kind of order zero ($ λ^{*} $≥ 2 j02 π > 8 π). Moreover, the family of solution uλ blows-up as λ → 8 π. © Springer Science + Business Media, Inc. 2006 |
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A blowing-up branch of solutions for a mean field equation |
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