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

Yunfeng Jia; Yi Li; Jianhua Wu; Hong-Kun Xu

doi:10.3934/dcds.2019227

Previous Article
Global stabilization of the full attraction-repulsion Keller-Segel system
DCDS Home
This Issue
Next Article
A stage structured model of delay differential equations for Aedes mosquito population suppression

June 2020, 40(6): 3485-3507. doi: 10.3934/dcds.2019227

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

Yunfeng Jia ^1, , Yi Li ^2,, , Jianhua Wu ^1, and Hong-Kun Xu ^3,

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.

Keywords: Cauchy problem, positive radial solutions, stability, sub- and super-solutions, Matukuma-type equation.

Mathematics Subject Classification: Primary: 35J10, 35J20; Secondary: 35J65.

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, 2020, 40 (6) : 3485-3507. 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

show all references

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

[1]	Haiyan Wang. Existence and nonexistence of positive radial solutions for quasilinear systems. Conference Publications, 2009, 2009 (Special) : 810-817. doi: 10.3934/proc.2009.2009.810
[2]	Ka Luen Cheung, Man Chun Leung. Asymptotic behavior of positive solutions of the equation $ \Delta u + K u^{\frac{n+2}{n-2}} = 0$ in $IR^n$ and positive scalar curvature. Conference Publications, 2001, 2001 (Special) : 109-120. doi: 10.3934/proc.2001.2001.109
[3]	Hildeberto E. Cabral, Zhihong Xia. Subharmonic solutions in the restricted three-body problem. Discrete & Continuous Dynamical Systems - A, 1995, 1 (4) : 463-474. doi: 10.3934/dcds.1995.1.463
[4]	Wentao Huang, Jianlin Xiang. Soliton solutions for a quasilinear Schrödinger equation with critical exponent. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1309-1333. doi: 10.3934/cpaa.2016.15.1309
[5]	Kin Ming Hui, Soojung Kim. Asymptotic large time behavior of singular solutions of the fast diffusion equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5943-5977. doi: 10.3934/dcds.2017258
[6]	Gloria Paoli, Gianpaolo Piscitelli, Rossanno Sannipoli. A stability result for the Steklov Laplacian Eigenvalue Problem with a spherical obstacle. Communications on Pure & Applied Analysis, 2021, 20 (1) : 145-158. doi: 10.3934/cpaa.2020261
[7]	Giovanni Cimatti. Forced periodic solutions for piezoelectric crystals. Communications on Pure & Applied Analysis, 2005, 4 (2) : 475-485. doi: 10.3934/cpaa.2005.4.475
[8]	Jaume Llibre, Luci Any Roberto. On the periodic solutions of a class of Duffing differential equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 277-282. doi: 10.3934/dcds.2013.33.277
[9]	Ian Schindler, Kyril Tintarev. Mountain pass solutions to semilinear problems with critical nonlinearity. Conference Publications, 2007, 2007 (Special) : 912-919. doi: 10.3934/proc.2007.2007.912
[10]	Shu-Yu Hsu. Existence and properties of ancient solutions of the Yamabe flow. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 91-129. doi: 10.3934/dcds.2018005
[11]	Jian Yang, Bendong Lou. Traveling wave solutions of competitive models with free boundaries. Discrete & Continuous Dynamical Systems - B, 2014, 19 (3) : 817-826. doi: 10.3934/dcdsb.2014.19.817
[12]	Palash Sarkar, Subhadip Singha. Verifying solutions to LWE with implications for concrete security. Advances in Mathematics of Communications, 2021, 15 (2) : 257-266. doi: 10.3934/amc.2020057
[13]	Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825
[14]	Haibo Cui, Haiyan Yin. Convergence rate of solutions toward stationary solutions to the isentropic micropolar fluid model in a half line. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020210
[15]	Yanqin Fang, Jihui Zhang. Multiplicity of solutions for the nonlinear Schrödinger-Maxwell system. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1267-1279. doi: 10.3934/cpaa.2011.10.1267
[16]	Yimin Zhang, Youjun Wang, Yaotian Shen. Solutions for quasilinear Schrödinger equations with critical Sobolev-Hardy exponents. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1037-1054. doi: 10.3934/cpaa.2011.10.1037
[17]	Elvise Berchio, Filippo Gazzola, Dario Pierotti. Nodal solutions to critical growth elliptic problems under Steklov boundary conditions. Communications on Pure & Applied Analysis, 2009, 8 (2) : 533-557. doi: 10.3934/cpaa.2009.8.533
[18]	Wei-Jian Bo, Guo Lin, Shigui Ruan. Traveling wave solutions for time periodic reaction-diffusion systems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4329-4351. doi: 10.3934/dcds.2018189
[19]	Luke Finlay, Vladimir Gaitsgory, Ivan Lebedev. Linear programming solutions of periodic optimization problems: approximation of the optimal control. Journal of Industrial & Management Optimization, 2007, 3 (2) : 399-413. doi: 10.3934/jimo.2007.3.399
[20]	A. Aghajani, S. F. Mottaghi. Regularity of extremal solutions of semilinaer fourth-order elliptic problems with general nonlinearities. Communications on Pure & Applied Analysis, 2018, 17 (3) : 887-898. doi: 10.3934/cpaa.2018044

American Institute of Mathematical Sciences

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

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

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

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors