# American Institute of Mathematical Sciences

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.
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)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.
 [1] Tohru Tsujikawa, Kousuke Kuto, Yasuhito Miyamoto, Hirofumi Izuhara. Stationary solutions for some shadow system of the Keller-Segel model with logistic growth. Discrete and Continuous Dynamical Systems - S, 2015, 8 (5) : 1023-1034. doi: 10.3934/dcdss.2015.8.1023 [2] Hao Yu, Wei Wang, Sining Zheng. Boundedness of solutions to a fully parabolic Keller-Segel system with nonlinear sensitivity. Discrete and Continuous Dynamical Systems - B, 2017, 22 (4) : 1635-1644. doi: 10.3934/dcdsb.2017078 [3] Hao Yu, Wei Wang, Sining Zheng. Global boundedness of solutions to a Keller-Segel system with nonlinear sensitivity. Discrete and Continuous Dynamical Systems - B, 2016, 21 (4) : 1317-1327. doi: 10.3934/dcdsb.2016.21.1317 [4] Hui Huang, Jian-Guo Liu. Well-posedness for the Keller-Segel equation with fractional Laplacian and the theory of propagation of chaos. Kinetic and Related Models, 2016, 9 (4) : 715-748. doi: 10.3934/krm.2016013 [5] Tobias Black. Global generalized solutions to a parabolic-elliptic Keller-Segel system with singular sensitivity. Discrete and Continuous Dynamical Systems - S, 2020, 13 (2) : 119-137. doi: 10.3934/dcdss.2020007 [6] Xinru Cao. Global bounded solutions of the higher-dimensional Keller-Segel system under smallness conditions in optimal spaces. Discrete and Continuous Dynamical Systems, 2015, 35 (5) : 1891-1904. doi: 10.3934/dcds.2015.35.1891 [7] Piotr Biler, Ignacio Guerra, Grzegorz Karch. Large global-in-time solutions of the parabolic-parabolic Keller-Segel system on the plane. Communications on Pure and Applied Analysis, 2015, 14 (6) : 2117-2126. doi: 10.3934/cpaa.2015.14.2117 [8] Johannes Lankeit. Infinite time blow-up of many solutions to a general quasilinear parabolic-elliptic Keller-Segel system. Discrete and Continuous Dynamical Systems - S, 2020, 13 (2) : 233-255. doi: 10.3934/dcdss.2020013 [9] J. Ignacio Tello. Radially symmetric solutions for a Keller-Segel system with flux limitation and nonlinear diffusion. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022045 [10] Ping Liu, Junping Shi, Zhi-An Wang. Pattern formation of the attraction-repulsion Keller-Segel system. Discrete and Continuous Dynamical Systems - B, 2013, 18 (10) : 2597-2625. doi: 10.3934/dcdsb.2013.18.2597 [11] Jinhuan Wang, Li Chen, Liang Hong. Parabolic elliptic type Keller-Segel system on the whole space case. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 1061-1084. doi: 10.3934/dcds.2016.36.1061 [12] Hongyun Peng, Zhi-An Wang, Kun Zhao, Changjiang Zhu. Boundary layers and stabilization of the singular Keller-Segel system. Kinetic and Related Models, 2018, 11 (5) : 1085-1123. doi: 10.3934/krm.2018042 [13] Hai-Yang Jin, Zhi-An Wang. Global stabilization of the full attraction-repulsion Keller-Segel system. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3509-3527. doi: 10.3934/dcds.2020027 [14] Qi Wang. Boundary spikes of a Keller-Segel chemotaxis system with saturated logarithmic sensitivity. Discrete and Continuous Dynamical Systems - B, 2015, 20 (4) : 1231-1250. doi: 10.3934/dcdsb.2015.20.1231 [15] Hai-Yang Jin, Zhi-An Wang. The Keller-Segel system with logistic growth and signal-dependent motility. Discrete and Continuous Dynamical Systems - B, 2021, 26 (6) : 3023-3041. doi: 10.3934/dcdsb.2020218 [16] Jaewook Ahn, Kyungkeun Kang. On a Keller-Segel system with logarithmic sensitivity and non-diffusive chemical. Discrete and Continuous Dynamical Systems, 2014, 34 (12) : 5165-5179. doi: 10.3934/dcds.2014.34.5165 [17] Kenneth H. Karlsen, Süleyman Ulusoy. On a hyperbolic Keller-Segel system with degenerate nonlinear fractional diffusion. Networks and Heterogeneous Media, 2016, 11 (1) : 181-201. doi: 10.3934/nhm.2016.11.181 [18] Miaoqing Tian, Sining Zheng. Global boundedness versus finite-time blow-up of solutions to a quasilinear fully parabolic Keller-Segel system of two species. Communications on Pure and Applied Analysis, 2016, 15 (1) : 243-260. doi: 10.3934/cpaa.2016.15.243 [19] Xiaoming Fu, Quentin Griette, Pierre Magal. Existence and uniqueness of solutions for a hyperbolic Keller–Segel equation. Discrete and Continuous Dynamical Systems - B, 2021, 26 (4) : 1931-1966. doi: 10.3934/dcdsb.2020326 [20] Jiahang Che, Li Chen, Simone GÖttlich, Anamika Pandey, Jing Wang. Boundary layer analysis from the Keller-Segel system to the aggregation system in one space dimension. Communications on Pure and Applied Analysis, 2017, 16 (3) : 1013-1036. doi: 10.3934/cpaa.2017049

2020 Impact Factor: 1.392