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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 |
$ \Delta u+\sum\limits_{i = 1}^{k}K_i(|x|)u^{p_i}+\mu f(|x|) = 0, \quad x\in\mathbb{R}^n, $ |
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. 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. |
[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). 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. |
[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. |
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. |
[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. 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. |
[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). 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. |
[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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