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Geodesic planes in geometrically finite manifolds
A general existence result for stationary solutions to the Keller-Segel system
Università degli Studi Roma Tre, Dipartimento di Matematica e Fisica, Largo S. Leonardo Murialdo 1, 00146 Roma, Italy |
$\left\{ \begin{gathered} - \Delta u + \beta u = \rho \left( {\frac{{{e^u}}}{{\int_\Omega {{e^u}} }} - \frac{1}{{\left| \Omega \right|}}} \right)\;\;\;\;\;\;{\text{in}}\;\Omega \hfill \\ {\partial _\nu }u = 0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\text{on}}\;\partial \Omega \hfill \\ \end{gathered} \right.,$ |
References:
[1] |
O. Agudelo and A. Pistoia, Boundary concentration phenomena for the higher-dimensional Keller-Segel system, Calc. Var. Partial Differential Equations, 55 (2016), Art. 132, 31pp.
doi: 10.1007/s00526-016-1083-7. |
[2] |
M. Ahmedou, S. Kallel and C. B. Ndiaye, The resonant boundary Q-curvature problem and boundary weighted barycenters, preprint. |
[3] |
D. Bartolucci, F. De Marchis and A. Malchiodi,
Supercritical conformal metrics on surfaces with conical singularities, Int. Math. Res. Not. IMRN, (2011), 5625-5643.
doi: 10.1093/imrn/rnq285. |
[4] |
L. Battaglia,
Existence and multiplicity result for the singular {T}oda system, J. Math. Anal. Appl., 424 (2015), 49-85.
doi: 10.1016/j.jmaa.2014.10.081. |
[5] |
L. Battaglia,
B2 and G2 Toda systems on compact surfaces: A variational approach,
Journal of Mathematical Physics, 58 (2017), 011506
doi: 10.1063/1.4974774. |
[6] |
L. Battaglia, A. Jevnikar, A. Malchiodi and D. Ruiz,
A general existence result for the Toda system on compact surfaces, Adv. Math., 285 (2015), 937-979.
doi: 10.1016/j.aim.2015.07.036. |
[7] |
L. Battaglia and A. Malchiodi,
Existence and non-existence results for the SU(3) singular Toda system on compact surfaces, J. Funct. Anal., 270 (2016), 3750-3807.
doi: 10.1016/j.jfa.2015.12.011. |
[8] |
L. Battaglia and G. Mancini,
A note on compactness properties of the singular Toda system, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 26 (2015), 299-307.
doi: 10.4171/RLM/708. |
[9] |
D. Bonheure, J.-B. Casteras and B. Noris,
Layered solutions with unbounded mass for the Keller-Segel equation, J. Fixed Point Theory Appl., 19 (2017), 529-558.
doi: 10.1007/s11784-016-0364-2. |
[10] |
D. Bonheure, J.-B. Casteras and B. Noris, Multiple positive solutions of the stationary Keller-Segel system,
Calc. Var. Partial Differential Equations, 56 (2017), Art. 74, 35pp.
doi: 10.1007/s00526-017-1163-3. |
[11] |
H. Brezis and F. Merle,
Uniform estimates and blow-up behavior for solutions of -Δu = V(x)eu in two dimensions, Comm. Partial Differential Equations, 16 (1991), 1223-1253.
doi: 10.1080/03605309108820797. |
[12] |
A. Carlotto and A. Malchiodi,
Weighted barycentric sets and singular Liouville equations on compact surfaces, J. Funct. Anal., 262 (2012), 409-450.
doi: 10.1016/j.jfa.2011.09.012. |
[13] |
S.-Y. A. Chang and P. C. Yang, Conformal deformation of metrics on S2, J. Differential Geom., 27 (1988), 259-296, URL http://projecteuclid.org/euclid.jdg/1214441783. |
[14] |
F. De Marchis,
Generic multiplicity for a scalar field equation on compact surfaces, J. Funct. Anal., 259 (2010), 2165-2192.
doi: 10.1016/j.jfa.2010.07.003. |
[15] |
F. De Marchis, R. López-Soriano and D. Ruiz,
Compactness, existence and multiplicity for the singular mean field problem with sign-changing potentials, J. Math. Pures Appl. (9), 115 (2018), 237-267.
doi: 10.1016/j.matpur.2017.11.007. |
[16] |
M. del Pino, A. Pistoia and G. Vaira,
Large mass boundary condensation patterns in the stationary Keller-Segel system, J. Differential Equations, 261 (2016), 3414-3462.
doi: 10.1016/j.jde.2016.05.032. |
[17] |
Z. Djadli and A. Malchiodi,
Existence of conformal metrics with constant Q-curvature, Ann. of Math. (2), 168 (2008), 813-858.
doi: 10.4007/annals.2008.168.813. |
[18] |
A. Hatcher,
Algebraic Topology, Cambridge University Press, Cambridge, 2002. |
[19] |
A. Jevnikar,
A note on a multiplicity result for the mean field equation on compact surfaces, Adv. Nonlinear Stud., 16 (2016), 221-229.
doi: 10.1515/ans-2015-5009. |
[20] |
A. Jevnikar, S. Kallel and A. Malchiodi,
A topological join construction and the Toda system on compact surfaces of arbitrary genus, Anal. PDE, 8 (2015), 1963-2027.
doi: 10.2140/apde.2015.8.1963. |
[21] |
S. Kallel and R. Karoui,
Symmetric joins and weighted barycenters, Adv. Nonlinear Stud., 11 (2011), 117-143.
doi: 10.1515/ans-2011-0106. |
[22] |
E. F. Keller and L. A. Segel,
Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415.
doi: 10.1016/0022-5193(70)90092-5. |
[23] |
M. Lucia,
A deformation lemma with an application to a mean field equation, Topol. Methods Nonlinear Anal., 30 (2007), 113-138.
|
[24] |
A. Malchiodi,
Topological methods for an elliptic equation with exponential nonlinearities, Discrete Contin. Dyn. Syst., 21 (2008), 277-294.
doi: 10.3934/dcds.2008.21.277. |
[25] |
A. Malchiodi,
Variational analysis of Toda systems, Chin. Ann. Math. Ser. B, 38 (2017), 539-562.
doi: 10.1007/s11401-017-1082-9. |
[26] |
A. Malchiodi,
A variational approach to Liouville equations, Boll. Unione Mat. Ital., 10 (2017), 75-97.
doi: 10.1007/s40574-016-0092-y. |
[27] |
J. Moser,
A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J., 20 (1970/71), 1077-1092.
doi: 10.1512/iumj.1971.20.20101. |
[28] |
C. B. Ndiaye,
Conformal metrics with constant Q-curvature for manifolds with boundary, Comm. Anal. Geom., 16 (2008), 1049-1124.
doi: 10.4310/CAG.2008.v16.n5.a6. |
[29] |
A. Pistoia and G. Vaira,
Steady states with unbounded mass of the Keller-Segel system, Proc. Roy. Soc. Edinburgh Sect. A, 145 (2015), 203-222.
doi: 10.1017/S0308210513000619. |
[30] |
G. Wang and J. Wei,
Steady state solutions of a reaction-diffusion system modeling chemotaxis, Math. Nachr., 233/234 (2002), 221-236.
doi: 10.1002/1522-2616(200201)233:1<221::AID-MANA221>3.3.CO;2-D. |
show all references
References:
[1] |
O. Agudelo and A. Pistoia, Boundary concentration phenomena for the higher-dimensional Keller-Segel system, Calc. Var. Partial Differential Equations, 55 (2016), Art. 132, 31pp.
doi: 10.1007/s00526-016-1083-7. |
[2] |
M. Ahmedou, S. Kallel and C. B. Ndiaye, The resonant boundary Q-curvature problem and boundary weighted barycenters, preprint. |
[3] |
D. Bartolucci, F. De Marchis and A. Malchiodi,
Supercritical conformal metrics on surfaces with conical singularities, Int. Math. Res. Not. IMRN, (2011), 5625-5643.
doi: 10.1093/imrn/rnq285. |
[4] |
L. Battaglia,
Existence and multiplicity result for the singular {T}oda system, J. Math. Anal. Appl., 424 (2015), 49-85.
doi: 10.1016/j.jmaa.2014.10.081. |
[5] |
L. Battaglia,
B2 and G2 Toda systems on compact surfaces: A variational approach,
Journal of Mathematical Physics, 58 (2017), 011506
doi: 10.1063/1.4974774. |
[6] |
L. Battaglia, A. Jevnikar, A. Malchiodi and D. Ruiz,
A general existence result for the Toda system on compact surfaces, Adv. Math., 285 (2015), 937-979.
doi: 10.1016/j.aim.2015.07.036. |
[7] |
L. Battaglia and A. Malchiodi,
Existence and non-existence results for the SU(3) singular Toda system on compact surfaces, J. Funct. Anal., 270 (2016), 3750-3807.
doi: 10.1016/j.jfa.2015.12.011. |
[8] |
L. Battaglia and G. Mancini,
A note on compactness properties of the singular Toda system, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 26 (2015), 299-307.
doi: 10.4171/RLM/708. |
[9] |
D. Bonheure, J.-B. Casteras and B. Noris,
Layered solutions with unbounded mass for the Keller-Segel equation, J. Fixed Point Theory Appl., 19 (2017), 529-558.
doi: 10.1007/s11784-016-0364-2. |
[10] |
D. Bonheure, J.-B. Casteras and B. Noris, Multiple positive solutions of the stationary Keller-Segel system,
Calc. Var. Partial Differential Equations, 56 (2017), Art. 74, 35pp.
doi: 10.1007/s00526-017-1163-3. |
[11] |
H. Brezis and F. Merle,
Uniform estimates and blow-up behavior for solutions of -Δu = V(x)eu in two dimensions, Comm. Partial Differential Equations, 16 (1991), 1223-1253.
doi: 10.1080/03605309108820797. |
[12] |
A. Carlotto and A. Malchiodi,
Weighted barycentric sets and singular Liouville equations on compact surfaces, J. Funct. Anal., 262 (2012), 409-450.
doi: 10.1016/j.jfa.2011.09.012. |
[13] |
S.-Y. A. Chang and P. C. Yang, Conformal deformation of metrics on S2, J. Differential Geom., 27 (1988), 259-296, URL http://projecteuclid.org/euclid.jdg/1214441783. |
[14] |
F. De Marchis,
Generic multiplicity for a scalar field equation on compact surfaces, J. Funct. Anal., 259 (2010), 2165-2192.
doi: 10.1016/j.jfa.2010.07.003. |
[15] |
F. De Marchis, R. López-Soriano and D. Ruiz,
Compactness, existence and multiplicity for the singular mean field problem with sign-changing potentials, J. Math. Pures Appl. (9), 115 (2018), 237-267.
doi: 10.1016/j.matpur.2017.11.007. |
[16] |
M. del Pino, A. Pistoia and G. Vaira,
Large mass boundary condensation patterns in the stationary Keller-Segel system, J. Differential Equations, 261 (2016), 3414-3462.
doi: 10.1016/j.jde.2016.05.032. |
[17] |
Z. Djadli and A. Malchiodi,
Existence of conformal metrics with constant Q-curvature, Ann. of Math. (2), 168 (2008), 813-858.
doi: 10.4007/annals.2008.168.813. |
[18] |
A. Hatcher,
Algebraic Topology, Cambridge University Press, Cambridge, 2002. |
[19] |
A. Jevnikar,
A note on a multiplicity result for the mean field equation on compact surfaces, Adv. Nonlinear Stud., 16 (2016), 221-229.
doi: 10.1515/ans-2015-5009. |
[20] |
A. Jevnikar, S. Kallel and A. Malchiodi,
A topological join construction and the Toda system on compact surfaces of arbitrary genus, Anal. PDE, 8 (2015), 1963-2027.
doi: 10.2140/apde.2015.8.1963. |
[21] |
S. Kallel and R. Karoui,
Symmetric joins and weighted barycenters, Adv. Nonlinear Stud., 11 (2011), 117-143.
doi: 10.1515/ans-2011-0106. |
[22] |
E. F. Keller and L. A. Segel,
Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415.
doi: 10.1016/0022-5193(70)90092-5. |
[23] |
M. Lucia,
A deformation lemma with an application to a mean field equation, Topol. Methods Nonlinear Anal., 30 (2007), 113-138.
|
[24] |
A. Malchiodi,
Topological methods for an elliptic equation with exponential nonlinearities, Discrete Contin. Dyn. Syst., 21 (2008), 277-294.
doi: 10.3934/dcds.2008.21.277. |
[25] |
A. Malchiodi,
Variational analysis of Toda systems, Chin. Ann. Math. Ser. B, 38 (2017), 539-562.
doi: 10.1007/s11401-017-1082-9. |
[26] |
A. Malchiodi,
A variational approach to Liouville equations, Boll. Unione Mat. Ital., 10 (2017), 75-97.
doi: 10.1007/s40574-016-0092-y. |
[27] |
J. Moser,
A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J., 20 (1970/71), 1077-1092.
doi: 10.1512/iumj.1971.20.20101. |
[28] |
C. B. Ndiaye,
Conformal metrics with constant Q-curvature for manifolds with boundary, Comm. Anal. Geom., 16 (2008), 1049-1124.
doi: 10.4310/CAG.2008.v16.n5.a6. |
[29] |
A. Pistoia and G. Vaira,
Steady states with unbounded mass of the Keller-Segel system, Proc. Roy. Soc. Edinburgh Sect. A, 145 (2015), 203-222.
doi: 10.1017/S0308210513000619. |
[30] |
G. Wang and J. Wei,
Steady state solutions of a reaction-diffusion system modeling chemotaxis, Math. Nachr., 233/234 (2002), 221-236.
doi: 10.1002/1522-2616(200201)233:1<221::AID-MANA221>3.3.CO;2-D. |
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