# American Institute of Mathematical Sciences

doi: 10.3934/dcds.2019227

## Cauchy problem of semilinear inhomogeneous elliptic equations of Matukuma-type with multiple growth terms

 1 School of Mathematics and Information Science, Shaanxi Normal University, Xi'an, Shaanxi 710062, China 2 Department of Mathematics and Computer Science, John Jay College of Criminal Justice, CUNY, New York, NY 10019, USA 3 School of Science, Hangzhou Dianzi University, Hangzhou, Zhejiang 310018, China

* Corresponding author: Yi Li

Received  April 2018 Revised  November 2018 Published  June 2019

Fund Project: Y. Jia and J. Wu are supported in part by the Natural Science Foundations of China (11771262, 11671243, 61672021), by the Natural Science Basic Research Plan in Shaanxi Province of China (2018JM1020)

We consider the structure and the stability of positive radial solutions of a semilinear inhomogeneous elliptic equation with multiple growth terms
 $\Delta u+\sum\limits_{i = 1}^{k}K_i(|x|)u^{p_i}+\mu f(|x|) = 0, \quad x\in\mathbb{R}^n,$
which is a generalization of Matukuma's equation describing the dynamics of a globular cluster of stars. Equations similar to this kind have come up both in geometry and in physics, and have been a subject of extensive studies. Our result shows that any positive radial solution is stable or weakly asymptotically stable with respect to certain norm.
Citation: Yunfeng Jia, Yi Li, Jianhua Wu, Hong-Kun Xu. Cauchy problem of semilinear inhomogeneous elliptic equations of Matukuma-type with multiple growth terms. Discrete & Continuous Dynamical Systems - A, doi: 10.3934/dcds.2019227
##### References:
 [1] D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. Math., 30 (1978), 33-76. doi: 10.1016/0001-8708(78)90130-5. Google Scholar [2] S. Bae, Infinite multiplicity and separation structure of positive solutions for a semilinear elliptic equation in $\mathbb{R}^n$, J. Differential Equations, 200 (2004), 274-311. doi: 10.1016/j.jde.2003.11.006. Google Scholar [3] S. Bae, Separation structure of positive radial solutions of a semilinear elliptic equation in $\mathbb{R}^n$, J. Differential Equations, 194 (2003), 460-499. doi: 10.1016/S0022-0396(03)00172-4. Google Scholar [4] S. Bae and T.-K. Chang, On a class of semilinear elliptic equations in $\mathbb{R}^n$, J. Differential Equations, 185 (2002), 225-250. doi: 10.1006/jdeq.2001.4162. Google Scholar [5] S. Bae and W.-M. Ni, Existence and infinite multiplicity for an inhomogeneous semilinear elliptic equation on $\mathbb{R}^n$, Math. Ann., 320 (2001), 191-210. doi: 10.1007/PL00004468. Google Scholar [6] S.-H. Chen and G.-Z. Lu, Asymptotic behavior of radial solutions for a class of semilinear elliptic equations, J. Differential Equations, 133 (1997), 340-354. doi: 10.1006/jdeq.1996.3208. Google Scholar [7] Y. Deng and Y. Li, Existence of multiple positive solutions for a semilinear elliptic equation, Adv. Differential Equations, 2 (1997), 361-382. Google Scholar [8] Y. Deng, Y. Li and Y. Liu, On the stability of the positive radial steady states for a semilinear Cauchy problem, Nonlinear Anal., 54 (2003), 291-318. doi: 10.1016/S0362-546X(03)00064-6. Google Scholar [9] Y. Deng, Y. Li and F. Yang, On the stability of the positive steady states for a nonhomogeneous semilinear Cauchy problem, J. Differential Equations, 228 (2006), 507-529. doi: 10.1016/j.jde.2006.02.010. Google Scholar [10] Y. Deng and F. Yang, Existence and asymptotic behavior of positive solutions for an inhomogeneous semilinear elliptic equation, Nonlinear Anal., 68 (2008), 246-272. doi: 10.1016/j.na.2006.10.046. Google Scholar [11] W.-R. Derrick, S. Chen and J. A. Cima, Oscillatory radial solutions of semilinear elliptic equations, J. Math. Anal. Appl., 208 (1997), 425-445. doi: 10.1006/jmaa.1997.5325. Google Scholar [12] W.-Y. Ding and W.-M. Ni, On the elliptic equation $\Delta u+ku^{\frac{(n+2)}{(n-2)}} = 0$ and related topics, Duke Math. J., 52 (1985), 485-506. doi: 10.1215/S0012-7094-85-05224-X. Google Scholar [13] A. S. Eddington, The dynamics of a globular stellar system, Monthly Notices Roy. Astronom. Soc., 75 (1915), 366-376. Google Scholar [14] M. Franca, Some results on the m-Laplace equations with two growth terms, J. Differential Equations, 17 (2005), 391-425. doi: 10.1007/s10884-005-4572-5. Google Scholar [15] A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, New Jersey, 1964. Google Scholar [16] H. Fujita, On the blowing up of solutions of the Cauchy problem for $u_t = \Delta u+u^{1+\alpha}$, J. Fac. Sci. Univ. Tokyo, 13 (1966), 109-124. Google Scholar [17] B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243. doi: 10.1007/BF01221125. Google Scholar [18] C.-F. Gui, On positive entire solutions of the elliptic equation $\Delta u+K(x)u^p = 0$ and its applications to Riemannian geometry, Proc. Roy. Soc. Edinburgh Sect. A, 126 (1996), 225-237. doi: 10.1017/S0308210500022708. Google Scholar [19] C.-F. Gui, Positive entire solutions of equation $\Delta u+f(x, u) = 0$, J. Differential Equations, 99 (1992), 245-280. doi: 10.1016/0022-0396(92)90023-G. Google Scholar [20] C.-F. Gui, W.-M. Ni and X.-F. Wang, Further study on a nonlinear heat equation, J. Differential Equations, 169 (2001), 588-613. doi: 10.1006/jdeq.2000.3909. Google Scholar [21] C.-F. Gui, W.-M. Ni and X.-F. Wang, On the stability and instability of positive steady state of a semilinear heat equation in $\mathbb{R}^n$, Comm. Pure Appl. Math., 45 (1992), 1153-1181. doi: 10.1002/cpa.3160450906. Google Scholar [22] N. Kawano, On bounded entire solutions of semilinear elliptic equations, Hiroshima Math. J., 14 (1984), 125-158. doi: 10.32917/hmj/1206133151. Google Scholar [23] T. Kusano and M. Naito, Oscillation theory of entire solutions of second order superlinear elliptic equations, Funkcial. Ekvac., 30 (1987), 269-282. Google Scholar [24] T.-Y. Lee and W.-M. Ni, Global existence, large time behavior and life span of solutions of semilinear parabolic Cauchy problems, Trans. Amer. Math. Soc., 333 (1992), 365-371. doi: 10.1090/S0002-9947-1992-1057781-6. Google Scholar [25] Y. Li, Asymptotic behavior of positive solutions of equation $\Delta u+K(x)u^p = 0$ in $\mathbb{R}^n$, J. Differential Equations, 95 (1992), 304-330. doi: 10.1016/0022-0396(92)90034-K. Google Scholar [26] Y. Li and W.-M. Ni, On the asymptotic behavior and radial symmetry of positive solutions of semilinear elliptic equations in $\mathbb{R}^n$ I. Asymptotic behavior, Arch. Rational Mech. Anal., 118 (1992), 195-222. doi: 10.1007/BF00387895. Google Scholar [27] Y. Liu, Y. Li and Y. Deng, Separation property of solution for a semilinear elliptic equation, J. Differential Equations, 163 (2000), 381-406. doi: 10.1006/jdeq.1999.3735. Google Scholar [28] T. Matukuma, Dynamics of globular clusters, Nippon Temmongakkai Yoho, 1 (1930), 68–89 (In Japanese).Google Scholar [29] M. Naito, Asymptotic behaviors of solutions of second order differential equations with integral coefficients, Trans. Amer. Math. Soc., 282 (1984), 577-588. doi: 10.1090/S0002-9947-1984-0732107-9. Google Scholar [30] M. Naito, A note on bounded positive entire solution of semiliner elliptic equations, Hiroshima Math. J., 14 (1984), 211-214. doi: 10.32917/hmj/1206133156. Google Scholar [31] W.-M. Ni, On the elliptic equation $\Delta u+K(x)u^{\frac{(n+2)}{(n-2)}}=0$, it's generalizations and applications in geometry, Indiana Univ. Math. J., 31 (1982), 493-529. doi: 10.1512/iumj.1982.31.31040. Google Scholar [32] W.-M. Ni and S. Yotsutani, Semilinear elliptic equations of Matukuma type and related topics, Japan J. Appl. Math., 5 (1988), 1-32. doi: 10.1007/BF03167899. Google Scholar [33] K. Nishihara, Asymptotic behaviors of solutions of second order differential equations, J. Math. Anal. Appl., 189 (1995), 424-441. doi: 10.1006/jmaa.1995.1028. Google Scholar [34] P. Polacik and E. Yanagida, A Liouville property and quasiconvergence for a semilinear heat equation, J. Differential Equations, 208 (2005), 194-214. doi: 10.1016/j.jde.2003.10.019. Google Scholar [35] P. Polacik and E. Yanagida, On bounded and unbounded global solutions of a supercritical semilinear heat equation, Math. Ann., 327 (2003), 745-771. doi: 10.1007/s00208-003-0469-y. Google Scholar [36] X.-F. Wang, On Cauchy problem for reaction-diffusion equations, Trans. Amer. Math. Soc., 337 (1993), 549-590. doi: 10.1090/S0002-9947-1993-1153016-5. Google Scholar [37] F. Weissler, Existence and nonexistence of global solution for semilinear heat equation, Israel J. Math., 38 (1981), 29-40. doi: 10.1007/BF02761845. Google Scholar [38] F. Yang, Entire positive solutions for an inhomogeneous semilinear biharmonic equation, Nonlinear Anal., 70 (2009), 1365-1376. doi: 10.1016/j.na.2008.02.016. Google Scholar [39] F. Yang and Z. Zhang, On the stability of the positive radial steady states for a semilinear Cauchy problem, Nonlinear Anal., 80 (2013), 109-121. doi: 10.1016/j.na.2012.11.003. Google Scholar [40] F. Yang and D. Zhang, Separation property of positive radial solutions for a general semilinear elliptic equation, Acta Math. Sci., 31 (2011), 181-193. doi: 10.1016/S0252-9602(11)60219-1. Google Scholar

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##### References:
 [1] D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. Math., 30 (1978), 33-76. doi: 10.1016/0001-8708(78)90130-5. Google Scholar [2] S. Bae, Infinite multiplicity and separation structure of positive solutions for a semilinear elliptic equation in $\mathbb{R}^n$, J. Differential Equations, 200 (2004), 274-311. doi: 10.1016/j.jde.2003.11.006. Google Scholar [3] S. Bae, Separation structure of positive radial solutions of a semilinear elliptic equation in $\mathbb{R}^n$, J. Differential Equations, 194 (2003), 460-499. doi: 10.1016/S0022-0396(03)00172-4. Google Scholar [4] S. Bae and T.-K. Chang, On a class of semilinear elliptic equations in $\mathbb{R}^n$, J. Differential Equations, 185 (2002), 225-250. doi: 10.1006/jdeq.2001.4162. Google Scholar [5] S. Bae and W.-M. Ni, Existence and infinite multiplicity for an inhomogeneous semilinear elliptic equation on $\mathbb{R}^n$, Math. Ann., 320 (2001), 191-210. doi: 10.1007/PL00004468. Google Scholar [6] S.-H. Chen and G.-Z. Lu, Asymptotic behavior of radial solutions for a class of semilinear elliptic equations, J. Differential Equations, 133 (1997), 340-354. doi: 10.1006/jdeq.1996.3208. Google Scholar [7] Y. Deng and Y. Li, Existence of multiple positive solutions for a semilinear elliptic equation, Adv. Differential Equations, 2 (1997), 361-382. Google Scholar [8] Y. Deng, Y. Li and Y. Liu, On the stability of the positive radial steady states for a semilinear Cauchy problem, Nonlinear Anal., 54 (2003), 291-318. doi: 10.1016/S0362-546X(03)00064-6. Google Scholar [9] Y. Deng, Y. Li and F. Yang, On the stability of the positive steady states for a nonhomogeneous semilinear Cauchy problem, J. Differential Equations, 228 (2006), 507-529. doi: 10.1016/j.jde.2006.02.010. Google Scholar [10] Y. Deng and F. Yang, Existence and asymptotic behavior of positive solutions for an inhomogeneous semilinear elliptic equation, Nonlinear Anal., 68 (2008), 246-272. doi: 10.1016/j.na.2006.10.046. Google Scholar [11] W.-R. Derrick, S. Chen and J. A. Cima, Oscillatory radial solutions of semilinear elliptic equations, J. Math. Anal. Appl., 208 (1997), 425-445. doi: 10.1006/jmaa.1997.5325. Google Scholar [12] W.-Y. Ding and W.-M. Ni, On the elliptic equation $\Delta u+ku^{\frac{(n+2)}{(n-2)}} = 0$ and related topics, Duke Math. J., 52 (1985), 485-506. doi: 10.1215/S0012-7094-85-05224-X. Google Scholar [13] A. S. Eddington, The dynamics of a globular stellar system, Monthly Notices Roy. Astronom. Soc., 75 (1915), 366-376. Google Scholar [14] M. Franca, Some results on the m-Laplace equations with two growth terms, J. Differential Equations, 17 (2005), 391-425. doi: 10.1007/s10884-005-4572-5. Google Scholar [15] A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, New Jersey, 1964. Google Scholar [16] H. Fujita, On the blowing up of solutions of the Cauchy problem for $u_t = \Delta u+u^{1+\alpha}$, J. Fac. Sci. Univ. Tokyo, 13 (1966), 109-124. Google Scholar [17] B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243. doi: 10.1007/BF01221125. Google Scholar [18] C.-F. Gui, On positive entire solutions of the elliptic equation $\Delta u+K(x)u^p = 0$ and its applications to Riemannian geometry, Proc. Roy. Soc. Edinburgh Sect. A, 126 (1996), 225-237. doi: 10.1017/S0308210500022708. Google Scholar [19] C.-F. Gui, Positive entire solutions of equation $\Delta u+f(x, u) = 0$, J. Differential Equations, 99 (1992), 245-280. doi: 10.1016/0022-0396(92)90023-G. Google Scholar [20] C.-F. Gui, W.-M. Ni and X.-F. Wang, Further study on a nonlinear heat equation, J. Differential Equations, 169 (2001), 588-613. doi: 10.1006/jdeq.2000.3909. Google Scholar [21] C.-F. Gui, W.-M. Ni and X.-F. Wang, On the stability and instability of positive steady state of a semilinear heat equation in $\mathbb{R}^n$, Comm. Pure Appl. Math., 45 (1992), 1153-1181. doi: 10.1002/cpa.3160450906. Google Scholar [22] N. Kawano, On bounded entire solutions of semilinear elliptic equations, Hiroshima Math. J., 14 (1984), 125-158. doi: 10.32917/hmj/1206133151. Google Scholar [23] T. Kusano and M. Naito, Oscillation theory of entire solutions of second order superlinear elliptic equations, Funkcial. Ekvac., 30 (1987), 269-282. Google Scholar [24] T.-Y. Lee and W.-M. Ni, Global existence, large time behavior and life span of solutions of semilinear parabolic Cauchy problems, Trans. Amer. Math. Soc., 333 (1992), 365-371. doi: 10.1090/S0002-9947-1992-1057781-6. Google Scholar [25] Y. Li, Asymptotic behavior of positive solutions of equation $\Delta u+K(x)u^p = 0$ in $\mathbb{R}^n$, J. Differential Equations, 95 (1992), 304-330. doi: 10.1016/0022-0396(92)90034-K. Google Scholar [26] Y. Li and W.-M. Ni, On the asymptotic behavior and radial symmetry of positive solutions of semilinear elliptic equations in $\mathbb{R}^n$ I. Asymptotic behavior, Arch. Rational Mech. Anal., 118 (1992), 195-222. doi: 10.1007/BF00387895. Google Scholar [27] Y. Liu, Y. Li and Y. Deng, Separation property of solution for a semilinear elliptic equation, J. Differential Equations, 163 (2000), 381-406. doi: 10.1006/jdeq.1999.3735. Google Scholar [28] T. Matukuma, Dynamics of globular clusters, Nippon Temmongakkai Yoho, 1 (1930), 68–89 (In Japanese).Google Scholar [29] M. Naito, Asymptotic behaviors of solutions of second order differential equations with integral coefficients, Trans. Amer. Math. Soc., 282 (1984), 577-588. doi: 10.1090/S0002-9947-1984-0732107-9. Google Scholar [30] M. Naito, A note on bounded positive entire solution of semiliner elliptic equations, Hiroshima Math. J., 14 (1984), 211-214. doi: 10.32917/hmj/1206133156. Google Scholar [31] W.-M. Ni, On the elliptic equation $\Delta u+K(x)u^{\frac{(n+2)}{(n-2)}}=0$, it's generalizations and applications in geometry, Indiana Univ. Math. J., 31 (1982), 493-529. doi: 10.1512/iumj.1982.31.31040. Google Scholar [32] W.-M. Ni and S. Yotsutani, Semilinear elliptic equations of Matukuma type and related topics, Japan J. Appl. Math., 5 (1988), 1-32. doi: 10.1007/BF03167899. Google Scholar [33] K. Nishihara, Asymptotic behaviors of solutions of second order differential equations, J. Math. Anal. Appl., 189 (1995), 424-441. doi: 10.1006/jmaa.1995.1028. Google Scholar [34] P. Polacik and E. Yanagida, A Liouville property and quasiconvergence for a semilinear heat equation, J. Differential Equations, 208 (2005), 194-214. doi: 10.1016/j.jde.2003.10.019. Google Scholar [35] P. Polacik and E. Yanagida, On bounded and unbounded global solutions of a supercritical semilinear heat equation, Math. Ann., 327 (2003), 745-771. doi: 10.1007/s00208-003-0469-y. Google Scholar [36] X.-F. Wang, On Cauchy problem for reaction-diffusion equations, Trans. Amer. Math. Soc., 337 (1993), 549-590. doi: 10.1090/S0002-9947-1993-1153016-5. Google Scholar [37] F. Weissler, Existence and nonexistence of global solution for semilinear heat equation, Israel J. Math., 38 (1981), 29-40. doi: 10.1007/BF02761845. Google Scholar [38] F. Yang, Entire positive solutions for an inhomogeneous semilinear biharmonic equation, Nonlinear Anal., 70 (2009), 1365-1376. doi: 10.1016/j.na.2008.02.016. Google Scholar [39] F. Yang and Z. Zhang, On the stability of the positive radial steady states for a semilinear Cauchy problem, Nonlinear Anal., 80 (2013), 109-121. doi: 10.1016/j.na.2012.11.003. Google Scholar [40] F. Yang and D. Zhang, Separation property of positive radial solutions for a general semilinear elliptic equation, Acta Math. Sci., 31 (2011), 181-193. doi: 10.1016/S0252-9602(11)60219-1. Google Scholar

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