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February  2019, 39(2): 905-926. doi: 10.3934/dcds.2019038

A general existence result for stationary solutions to the Keller-Segel system

Luca Battaglia

Università degli Studi Roma Tre, Dipartimento di Matematica e Fisica, Largo S. Leonardo Murialdo 1, 00146 Roma, Italy

Received  February 2018 Revised  August 2018 Published  November 2018

We consider the following Liouville-type PDE, which is related to stationary solutions of the Keller-Segel's model for chemotaxis:
$\left\{ \begin{gathered} - \Delta u + \beta u = \rho \left( {\frac{{{e^u}}}{{\int_\Omega {{e^u}} }} - \frac{1}{{\left| \Omega \right|}}} \right)\;\;\;\;\;\;{\text{in}}\;\Omega \hfill \\ {\partial _\nu }u = 0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\text{on}}\;\partial \Omega \hfill \\ \end{gathered} \right.,$
where $\Omega \subset {\mathbb{R}^2}$ is a smooth bounded domain and $\beta, ρ$ are real parameters. We prove existence of solutions under some algebraic conditions involving $\beta, ρ$. In particular, if $\Omega$ is not simply connected, then we can find solution for a generic choice of the parameters. We use variational and Morse-theoretical methods.
Keywords: Keller-Segel system, stationary solutions, Liouville equation, variational methods, Morse theory.
Mathematics Subject Classification: Primary: 35J61; Secondary: 35J20.
Citation: Luca Battaglia. A general existence result for stationary solutions to the Keller-Segel system. Discrete & Continuous Dynamical Systems, 2019, 39 (2) : 905-926. doi: 10.3934/dcds.2019038
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.  Google Scholar

[2]

M. Ahmedou, S. Kallel and C. B. Ndiaye, The resonant boundary Q-curvature problem and boundary weighted barycenters, preprint. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

A. Hatcher, Algebraic Topology, Cambridge University Press, Cambridge, 2002.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

S. Kallel and R. Karoui, Symmetric joins and weighted barycenters, Adv. Nonlinear Stud., 11 (2011), 117-143.  doi: 10.1515/ans-2011-0106.  Google Scholar

[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.  Google Scholar

[23]

M. Lucia, A deformation lemma with an application to a mean field equation, Topol. Methods Nonlinear Anal., 30 (2007), 113-138.   Google Scholar

[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.  Google Scholar

[25]

A. Malchiodi, Variational analysis of Toda systems, Chin. Ann. Math. Ser. B, 38 (2017), 539-562.  doi: 10.1007/s11401-017-1082-9.  Google Scholar

[26]

A. Malchiodi, A variational approach to Liouville equations, Boll. Unione Mat. Ital., 10 (2017), 75-97.  doi: 10.1007/s40574-016-0092-y.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

M. Ahmedou, S. Kallel and C. B. Ndiaye, The resonant boundary Q-curvature problem and boundary weighted barycenters, preprint. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

A. Hatcher, Algebraic Topology, Cambridge University Press, Cambridge, 2002.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

S. Kallel and R. Karoui, Symmetric joins and weighted barycenters, Adv. Nonlinear Stud., 11 (2011), 117-143.  doi: 10.1515/ans-2011-0106.  Google Scholar

[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.  Google Scholar

[23]

M. Lucia, A deformation lemma with an application to a mean field equation, Topol. Methods Nonlinear Anal., 30 (2007), 113-138.   Google Scholar

[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.  Google Scholar

[25]

A. Malchiodi, Variational analysis of Toda systems, Chin. Ann. Math. Ser. B, 38 (2017), 539-562.  doi: 10.1007/s11401-017-1082-9.  Google Scholar

[26]

A. Malchiodi, A variational approach to Liouville equations, Boll. Unione Mat. Ital., 10 (2017), 75-97.  doi: 10.1007/s40574-016-0092-y.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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