# 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, California State University, Northridge, CA 91325, 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. [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. [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. [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. [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. [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. [7] Y. Deng and Y. Li, Existence of multiple positive solutions for a semilinear elliptic equation, Adv. Differential Equations, 2 (1997), 361-382. [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. [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. [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. [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. [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. [13] A. S. Eddington, The dynamics of a globular stellar system, Monthly Notices Roy. Astronom. Soc., 75 (1915), 366-376. [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. [15] A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, New Jersey, 1964. [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. [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. [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. [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. [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. [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. [22] N. Kawano, On bounded entire solutions of semilinear elliptic equations, Hiroshima Math. J., 14 (1984), 125-158. doi: 10.32917/hmj/1206133151. [23] T. Kusano and M. Naito, Oscillation theory of entire solutions of second order superlinear elliptic equations, Funkcial. Ekvac., 30 (1987), 269-282. [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. [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. [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. [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. [28] T. Matukuma, Dynamics of globular clusters, Nippon Temmongakkai Yoho, 1 (1930), 68–89 (In Japanese). [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. [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. [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. [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. [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. [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. [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. [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. [37] F. Weissler, Existence and nonexistence of global solution for semilinear heat equation, Israel J. Math., 38 (1981), 29-40. doi: 10.1007/BF02761845. [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. [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. [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.

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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. [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. [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. [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. [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. [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. [7] Y. Deng and Y. Li, Existence of multiple positive solutions for a semilinear elliptic equation, Adv. Differential Equations, 2 (1997), 361-382. [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. [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. [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. [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. [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. [13] A. S. Eddington, The dynamics of a globular stellar system, Monthly Notices Roy. Astronom. Soc., 75 (1915), 366-376. [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. [15] A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, New Jersey, 1964. [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. [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. [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. [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. [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. [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. [22] N. Kawano, On bounded entire solutions of semilinear elliptic equations, Hiroshima Math. J., 14 (1984), 125-158. doi: 10.32917/hmj/1206133151. [23] T. Kusano and M. Naito, Oscillation theory of entire solutions of second order superlinear elliptic equations, Funkcial. Ekvac., 30 (1987), 269-282. [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. [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. [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. [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. [28] T. Matukuma, Dynamics of globular clusters, Nippon Temmongakkai Yoho, 1 (1930), 68–89 (In Japanese). [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. [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. [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. [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. [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. [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. [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. [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. [37] F. Weissler, Existence and nonexistence of global solution for semilinear heat equation, Israel J. Math., 38 (1981), 29-40. doi: 10.1007/BF02761845. [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. [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. [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.
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