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

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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

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 and 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.
[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)e^u 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)e^u 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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