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August 2017, 37(8): 4489-4505. doi: 10.3934/dcds.2017192

## Existence, nonexistence and uniqueness of positive stationary solutions of a singular Gierer-Meinhardt system

 1 School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou 221116, China 2 Department of Mathematics, School of Mathematics, Tianjin University, Tianjin 300072, China 3 School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou 221116, China

Received  November 2016 Revised  March 2017 Published  April 2017

This paper is concerned with the stationary Gierer-Meinhardt system with singularity:
 $\left\{\begin{array}{ll} d_1\Delta u-a_1 u+\frac{u^p}{v^q}+\rho_1(x)=0, \ \ & x\in\Omega, \\ d_2\Delta v-a_2 v+\frac{u^r}{v^s}+\rho_2(x)=0,\ \ & x\in\Omega,\\ u(x)>0,\ \ v(x)>0,\ \ & x\in \Omega,\\ \displaystyle u(x)=v(x)=0,\ \ & x\in\partial\Omega, \end{array}\right.$
where $-\infty < p < 1$, $-1 < s$, and $q, r, d_1, d_2$ are positive constants, $a_1, \, a_2$ are nonnegative constants, $\rho_1, \, \rho_2$ are smooth nonnegative functions and $\Omega\subset \mathbb{R}^d\, (d\geq1)$ is a bounded smooth domain. New sufficient conditions, some of which are necessary, on the existence of classical solutions are established. A uniqueness result of solutions in any space dimension is also derived. Previous results are substantially improved; moreover, a much simpler mathematical approach with potential application in other problems is developed.
Citation: Rui Peng, Xianfa Song, Lei Wei. Existence, nonexistence and uniqueness of positive stationary solutions of a singular Gierer-Meinhardt system. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4489-4505. doi: 10.3934/dcds.2017192
##### References:
 [1] S. Chen, Y. Salmaniw and R. Xu, Global existence for a singular Gierer-Meinhardt system, J. Differential Equations, 262 (2017), 2940-2960. doi: 10.1016/j.jde.2016.11.022. [2] S. Chen, Steady state solutions for a general activator-inhibitor model, Nonlinear Anal., 135 (2016), 84-96. doi: 10.1016/j.na.2016.01.013. [3] Y. S. Choi and P. J. McKenna, A singular Gierer-Meinhardt system of elliptic equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 17 (2000), 503-522. doi: 10.1016/S0294-1449(00)00115-3. [4] Y. S. Choi and P. J. McKenna, A singular Gierer-Meinhardt system of elliptic equations: The classical case, Nonlinear Anal., 55 (2003), 521-541. doi: 10.1016/j.na.2003.07.003. [5] M. Ghergu, Steady-state solutions for Gierer-Meinhardt type systems with Dirichlet boundary condition, Trans. Amer. Math. Soc., 361 (2009), 3953-3976. doi: 10.1090/S0002-9947-09-04670-4. [6] M. Ghergu, Lane-Emden systems with negative exponents, J. Funct. Anal., 258 (2010), 3295-3318. doi: 10.1016/j.jfa.2010.02.003. [7] M. Ghergu and V. Rădulescu, On a class of sublinear singular elliptic problems with convection term, J. Math. Anal. Appl., 311 (2005), 635-646. doi: 10.1016/j.jmaa.2005.03.012. [8] M. Ghergu and V. Rădulescu, On a class of singular Gierer-Meinhardt systems arising in morphogenesis, C. R. Math. Acad. Sci. Paris, 344 (2007), 163-168. doi: 10.1016/j.crma.2006.12.008. [9] M. Ghergu and V. Rădulescu, A singular Gierer-Meinhardt system with different source terms, Proc. Roy. Soc. Edinburgh Sect. A, 138 (2008), 1215-1234. doi: 10.1017/S0308210507000637. [10] M. Ghergu and V. Rădulescu, The influence of the distance function in some singular elliptic problems, Potential theory and stochastics in Albac, 125-137, Theta Ser. Adv. Math. , 11, Theta, Bucharest, 2009. [11] M. Ghergu and V. Rădulescu, Nonlinear PDEs. Mathematical Models in Biology, Chemistry and Population Genetics, With a foreword by Viorel Barbu. Springer Monographs in Mathematics. Springer, Heidelberg, 2012. xviii+391 pp. ISBN: 978-3-642-22663-2. doi: 10.1007/978-3-642-22664-9. [12] A. Gierer and H. Meinhardt, A theory of biological pattern formation, Kybernetik, 12 (1972), 30-39. doi: 10.1007/BF00289234. [13] H. Jiang, Global existence of solutions of an activator-inhibitor system, Discrete Contin. Dyn. Syst., 14 (2006), 737-751. doi: 10.3934/dcds.2006.14.737. [14] H. Jiang and W. -M. Ni, A priori estimates of stationary solutions of an activator-inhibitor system, Indiana Univ. Math. J., 56 (2007), 681-732. doi: 10.1512/iumj.2007.56.2982. [15] E. H. Kim, Singular Gierer-Meinhardt systems of elliptic boundary value problems, J. Math. Anal. Appl., 308 (2005), 1-10. doi: 10.1016/j.jmaa.2004.10.039. [16] E. H. Kim, A class of singular Gierer-Meinhardt systems of elliptic boundary value problems, Nonlinear Anal., 59 (2004), 305-318. doi: 10.1016/S0362-546X(04)00260-3. [17] A. Lazer and J. P. McKenna, On a singular nonlinear elliptic boundary value problem, Proc. Amer. Math. Soc., 111 (1991), 721-730. doi: 10.1090/S0002-9939-1991-1037213-9. [18] F. Li, R. Peng and X. F. Song, Global existence and finite time blow-up of solutions of a Gierer-Meinhardt system, J. Differential Equations, 262 (2017), 559-589. doi: 10.1016/j.jde.2016.09.040. [19] W. -M. Ni, Diffusion, cross-diffusion, and their spike-layer steady states, Notices Amer. Math. Soc., 45 (1998), 9-18. [20] W. -M. Ni, The Mathematics of Diffusion, CBMS-NSF Regional Conf. Ser. in Appl. Math. 82, SIAM, Philadelphia, 2011. doi: 10.1137/1.9781611971972. [21] W. -M. Ni, K. Suzuki and I. Takagi, The dynamics of a kynetics activator-inhibitor system, J. Differential Equations, 229 (2006), 426-465. doi: 10.1016/j.jde.2006.03.011. [22] W. -M. Ni, I. Takagi and E. Yanagida, Stability of least energy patterns of the shadow system for an activator-inhibitor model, Japan J. Indust. Appl. Math., 18 (2001), 259-272. doi: 10.1007/BF03168574. [23] W. -M. Ni and I. Takagi, On the Neumann problem for some semilinear elliptic equations and systems of activator-inhibitor type, Trans. Amer. Math. Soc., 297 (1986), 351-368. doi: 10.1090/S0002-9947-1986-0849484-2. [24] W. -M. Ni and I. Takagi, Point condensation generated by a reaction-diffusion system in axially symmetric domains, Japan J. Indust. Appl. Math., 12 (1995), 327-365. doi: 10.1007/BF03167294. [25] W. -M. Ni and J. Wei, On positive solutions concentrating on spheres for the Gierer-Meinhardt system, J. Differential Equations, 221 (2006), 158-189. doi: 10.1016/j.jde.2005.03.004. [26] C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992. [27] V. Rădulescu, Bifurcation and asymptotics for elliptic problems with singular nonlinearity. Elliptic and Parabolic Problems, 389-401, Progr. Nonlinear Differential Equations Appl. , 63, Birkhäuser, Basel, 2005. doi: 10.1007/3-7643-7384-9_38. [28] A. Trembley, M'emoires pour servir à l'histoire dun genre de polype d'eau douce, à bras en forme de corne, Verbeek, Leiden, Netherland, 1744. [29] A. M. Turing, The chemical basis of morphogenesis, Philosophical Transactions of the Royal Society (B), 237 (1952), 37-72. doi: 10.1098/rstb.1952.0012. [30] J. C. Wei and M. Winter, Mathematical Aspects of Pattern Formation in Biological Systems, Applied Mathematical Sciences, 189. Springer, London, 2014. xii+319 pp. ISBN: 978-1-4471-5525-6; 978-1-4471-5526-3. doi: 10.1007/978-1-4471-5526-3. [31] Q. X. Ye, Z. Y. Li, M. X. Wang and Y. P. Wu, Introduction to Reaction-Diffusion Equations, Second Edition, Science Press, Beijing, 2011. [32] Y. Zhang, Positive solutions of singular sublinear Dirichlet boundary value problems, SIAM J. Math. Anal., 26 (1995), 329-339. doi: 10.1137/S0036141093246087.

show all references

##### References:
 [1] S. Chen, Y. Salmaniw and R. Xu, Global existence for a singular Gierer-Meinhardt system, J. Differential Equations, 262 (2017), 2940-2960. doi: 10.1016/j.jde.2016.11.022. [2] S. Chen, Steady state solutions for a general activator-inhibitor model, Nonlinear Anal., 135 (2016), 84-96. doi: 10.1016/j.na.2016.01.013. [3] Y. S. Choi and P. J. McKenna, A singular Gierer-Meinhardt system of elliptic equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 17 (2000), 503-522. doi: 10.1016/S0294-1449(00)00115-3. [4] Y. S. Choi and P. J. McKenna, A singular Gierer-Meinhardt system of elliptic equations: The classical case, Nonlinear Anal., 55 (2003), 521-541. doi: 10.1016/j.na.2003.07.003. [5] M. Ghergu, Steady-state solutions for Gierer-Meinhardt type systems with Dirichlet boundary condition, Trans. Amer. Math. Soc., 361 (2009), 3953-3976. doi: 10.1090/S0002-9947-09-04670-4. [6] M. Ghergu, Lane-Emden systems with negative exponents, J. Funct. Anal., 258 (2010), 3295-3318. doi: 10.1016/j.jfa.2010.02.003. [7] M. Ghergu and V. Rădulescu, On a class of sublinear singular elliptic problems with convection term, J. Math. Anal. Appl., 311 (2005), 635-646. doi: 10.1016/j.jmaa.2005.03.012. [8] M. Ghergu and V. Rădulescu, On a class of singular Gierer-Meinhardt systems arising in morphogenesis, C. R. Math. Acad. Sci. Paris, 344 (2007), 163-168. doi: 10.1016/j.crma.2006.12.008. [9] M. Ghergu and V. Rădulescu, A singular Gierer-Meinhardt system with different source terms, Proc. Roy. Soc. Edinburgh Sect. A, 138 (2008), 1215-1234. doi: 10.1017/S0308210507000637. [10] M. Ghergu and V. Rădulescu, The influence of the distance function in some singular elliptic problems, Potential theory and stochastics in Albac, 125-137, Theta Ser. Adv. Math. , 11, Theta, Bucharest, 2009. [11] M. Ghergu and V. Rădulescu, Nonlinear PDEs. Mathematical Models in Biology, Chemistry and Population Genetics, With a foreword by Viorel Barbu. Springer Monographs in Mathematics. Springer, Heidelberg, 2012. xviii+391 pp. ISBN: 978-3-642-22663-2. doi: 10.1007/978-3-642-22664-9. [12] A. Gierer and H. Meinhardt, A theory of biological pattern formation, Kybernetik, 12 (1972), 30-39. doi: 10.1007/BF00289234. [13] H. Jiang, Global existence of solutions of an activator-inhibitor system, Discrete Contin. Dyn. Syst., 14 (2006), 737-751. doi: 10.3934/dcds.2006.14.737. [14] H. Jiang and W. -M. Ni, A priori estimates of stationary solutions of an activator-inhibitor system, Indiana Univ. Math. J., 56 (2007), 681-732. doi: 10.1512/iumj.2007.56.2982. [15] E. H. Kim, Singular Gierer-Meinhardt systems of elliptic boundary value problems, J. Math. Anal. Appl., 308 (2005), 1-10. doi: 10.1016/j.jmaa.2004.10.039. [16] E. H. Kim, A class of singular Gierer-Meinhardt systems of elliptic boundary value problems, Nonlinear Anal., 59 (2004), 305-318. doi: 10.1016/S0362-546X(04)00260-3. [17] A. Lazer and J. P. McKenna, On a singular nonlinear elliptic boundary value problem, Proc. Amer. Math. Soc., 111 (1991), 721-730. doi: 10.1090/S0002-9939-1991-1037213-9. [18] F. Li, R. Peng and X. F. Song, Global existence and finite time blow-up of solutions of a Gierer-Meinhardt system, J. Differential Equations, 262 (2017), 559-589. doi: 10.1016/j.jde.2016.09.040. [19] W. -M. Ni, Diffusion, cross-diffusion, and their spike-layer steady states, Notices Amer. Math. Soc., 45 (1998), 9-18. [20] W. -M. Ni, The Mathematics of Diffusion, CBMS-NSF Regional Conf. Ser. in Appl. Math. 82, SIAM, Philadelphia, 2011. doi: 10.1137/1.9781611971972. [21] W. -M. Ni, K. Suzuki and I. Takagi, The dynamics of a kynetics activator-inhibitor system, J. Differential Equations, 229 (2006), 426-465. doi: 10.1016/j.jde.2006.03.011. [22] W. -M. Ni, I. Takagi and E. Yanagida, Stability of least energy patterns of the shadow system for an activator-inhibitor model, Japan J. Indust. Appl. Math., 18 (2001), 259-272. doi: 10.1007/BF03168574. [23] W. -M. Ni and I. Takagi, On the Neumann problem for some semilinear elliptic equations and systems of activator-inhibitor type, Trans. Amer. Math. Soc., 297 (1986), 351-368. doi: 10.1090/S0002-9947-1986-0849484-2. [24] W. -M. Ni and I. Takagi, Point condensation generated by a reaction-diffusion system in axially symmetric domains, Japan J. Indust. Appl. Math., 12 (1995), 327-365. doi: 10.1007/BF03167294. [25] W. -M. Ni and J. Wei, On positive solutions concentrating on spheres for the Gierer-Meinhardt system, J. Differential Equations, 221 (2006), 158-189. doi: 10.1016/j.jde.2005.03.004. [26] C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992. [27] V. Rădulescu, Bifurcation and asymptotics for elliptic problems with singular nonlinearity. Elliptic and Parabolic Problems, 389-401, Progr. Nonlinear Differential Equations Appl. , 63, Birkhäuser, Basel, 2005. doi: 10.1007/3-7643-7384-9_38. [28] A. Trembley, M'emoires pour servir à l'histoire dun genre de polype d'eau douce, à bras en forme de corne, Verbeek, Leiden, Netherland, 1744. [29] A. M. Turing, The chemical basis of morphogenesis, Philosophical Transactions of the Royal Society (B), 237 (1952), 37-72. doi: 10.1098/rstb.1952.0012. [30] J. C. Wei and M. Winter, Mathematical Aspects of Pattern Formation in Biological Systems, Applied Mathematical Sciences, 189. Springer, London, 2014. xii+319 pp. ISBN: 978-1-4471-5525-6; 978-1-4471-5526-3. doi: 10.1007/978-1-4471-5526-3. [31] Q. X. Ye, Z. Y. Li, M. X. Wang and Y. P. Wu, Introduction to Reaction-Diffusion Equations, Second Edition, Science Press, Beijing, 2011. [32] Y. Zhang, Positive solutions of singular sublinear Dirichlet boundary value problems, SIAM J. Math. Anal., 26 (1995), 329-339. doi: 10.1137/S0036141093246087.
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