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July  2019, 39(7): 4127-4136. doi: 10.3934/dcds.2019166

## Stability and separation property of radial solutions to semilinear elliptic equations

 Department of Mathematics, Tokyo Institute of Technology, Tokyo 152-8551, Japan

Received  October 2018 Revised  January 2019 Published  April 2019

We study stability and separation property of solutions to Hénontype equations. In particular, assuming separation property of radial solutions, we shall show the stability of solutions. Moreover, we shall also study those properties of solutions to generalized Eddington equations.

Citation: Shoichi Hasegawa. Stability and separation property of radial solutions to semilinear elliptic equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 4127-4136. doi: 10.3934/dcds.2019166
##### References:
 [1] M. Badiale and G. Tarantello, A Sobolev-Hardy inequality with applications to a nonlinear elliptic equation arising in astrophysics, Arch. Ration. Mech. Anal., 163 (2002), 259-293. doi: 10.1007/s002050200201. Google Scholar [2] 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 [3] 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 [4] S. Bae, Positive entire solutions of semilinear elliptic equations with quadratically vanishing coefficient, J. Differential Equations, 237 (2007), 159-197. doi: 10.1016/j.jde.2007.03.003. Google Scholar [5] S. Bae, Entire solutions with asymptotic self-similarity for elliptic equations with exponential nonlinearity, J. Math. Anal. Appl., 428 (2015), 1085-1116. doi: 10.1016/j.jmaa.2015.03.036. Google Scholar [6] 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 [7] S. Bae and Y. Naito, Existence and separation of positive radial solutions for semilinear elliptic equations, J. Differential Equations, 257 (2014), 2430-2463. doi: 10.1016/j.jde.2014.05.042. Google Scholar [8] S. Bae and Y. Naito, Separation structure of radial solutions for semilinear elliptic equations with exponential nonlinearity, Discrete Contin. Dyn. Syst., 38 (2018), 4537-4554. doi: 10.3934/dcds.2018198. Google Scholar [9] C. Chen and H. Wang, Liouville theorems for the weighted Lane-Emden equation with finite Morse indices, Math. Methods Appl. Sci., 40 (2017), 4674-4682. Google Scholar [10] E. N. Dancer, Y. Du and Z. Guo, Finite Morse index solutions of an elliptic equation with supercritical exponent, J. Differential Equations, 250 (2011), 3281-3310. doi: 10.1016/j.jde.2011.02.005. Google Scholar [11] Y. Deng, Y. Li and F. Yang, On the positive radial solutions of a class of singular semilinear elliptic equations, J. Differential Equations, 253 (2012), 481-501. doi: 10.1016/j.jde.2012.02.017. Google Scholar [12] L. Dupaigne and A. Farina, Stable solutions of $-\Delta u = f(u)$ in $\mathbb{R}^{N}$, J. Eur. Math. Soc. (JEMS), 12 (2010), 855-882. doi: 10.4171/JEMS/217. Google Scholar [13] A. Farina, On the classification of solutions of the Lane-Emden equation on unbounded domains of $\mathbb{R}^{N}$, J. Math. Pures Appl. (9), 87 (2007), 537-561. doi: 10.1016/j.matpur.2007.03.001. Google Scholar [14] A. Farina, Stable solutions of $-\Delta u = e^{u}$ on $\mathbb{R}^{N}$, C. R. Math. Acad. Sci. Paris, 345 (2007), 63-66. doi: 10.1016/j.crma.2007.05.021. Google Scholar [15] A. Farina, Some symmetry results and Liouville-type theorems for solutions to semilinear equations, Nonlinear Anal., 121 (2015), 223-229. doi: 10.1016/j.na.2015.02.004. Google Scholar [16] A. Farina, B. Sciunzi and E. Valdinoci, Bernstein and De Giorgi type problems: New results via a geometric approach, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 7 (2008), 741-791. Google Scholar [17] C. Gui, Positive entire solutions of the equation $\Delta u+f(x,u) = 0$, J. Differential Equations, 99 (1992), 245-280. doi: 10.1016/0022-0396(92)90023-G. Google Scholar [18] C. Gui, K.-M. Ni and X. Wang, On the stability and instability of positive steady states of a semilinear heat equation in $\mathbb{R}^{n}$, Comm. Pure Appl. Math., 45 (1992), 1153-1181. doi: 10.1002/cpa.3160450906. Google Scholar [19] C. Gui, K.-M. Ni and X. Wang, Further study on a nonlinear heat equation, J. Differential Equations, 169 (2001), 588-613. doi: 10.1006/jdeq.2000.3909. Google Scholar [20] H. Hajlaoui, A. Harrabi and F. Mtiri, Liouville theorems for stable solutions of the weighted Lane-Emden system, Discrete Contin. Dyn. Syst., 37 (2017), 265-279. doi: 10.3934/dcds.2017011. Google Scholar [21] D. D. Joseph and T. S. Lundgren, Quasilinear Dirichlet problems driven by positive sources, Arch. Rational Mech. Anal., 49 (1972/73), 241-269. doi: 10.1007/BF00250508. Google Scholar [22] 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 [23] Y. Li, On the positive solutions of the Matukuma equation, Duke Math. J., 70 (1993), 575-589. doi: 10.1215/S0012-7094-93-07012-3. Google Scholar [24] Y. Liu, Y. Li and Y. Deng, Separation property of solutions for a semilinear elliptic equation, J. Differential Equations, 163 (2000), 381-406. doi: 10.1006/jdeq.1999.3735. Google Scholar [25] Y. Miyamoto, Intersection properties of radial solutions and global bifurcation diagrams for supercritical quasilinear elliptic equations, NoDEA Nonlinear Differential Equations Appl., 23 (2016), Art. 16, 24 pp. doi: 10.1007/s00030-016-0359-0. Google Scholar [26] Y. Miyamoto and K. Takahashi, Generalized Joseph-Lundgren exponent and intersection properties for supercritical quasilinear elliptic equations, Arch. Math. (Basel), 108 (2017), 71-83. doi: 10.1007/s00013-016-0969-0. Google Scholar [27] W.-M. Ni and S. Yotsutani, On Matukuma's equation and related topics, Proc. Japan Acad. Ser. A Math. Sci., 62 (1986), 260-263. doi: 10.3792/pjaa.62.260. Google Scholar [28] 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 [29] J. I. Tello, Stability of steady states of the Cauchy problem for the exponential reaction-diffusion equation, J. Math. Anal. Appl., 324 (2006), 381-396. doi: 10.1016/j.jmaa.2005.12.011. Google Scholar [30] S. Villegas, Asymptotic behavior of stable radial solutions of semilinear elliptic equations in $\mathbb{R}^{N}$, J. Math. Pures Appl. (9), 88 (2007), 241-250. doi: 10.1016/j.matpur.2007.06.004. Google Scholar [31] X. Wang, On the Cauchy problem for reaction-diffusion equations, Trans. Amer. Math. Soc., 337 (1993), 549-590. doi: 10.1090/S0002-9947-1993-1153016-5. Google Scholar [32] E. Yanagida, Structure of radial solutions to $\Delta u+K(|x|)|u|^{p-1}u=0$ in $\mathbb{R}^{N}$, SIAM J. Math. Anal., 27 (1996), 997-1014. doi: 10.1137/0527053. Google Scholar

show all references

##### References:
 [1] M. Badiale and G. Tarantello, A Sobolev-Hardy inequality with applications to a nonlinear elliptic equation arising in astrophysics, Arch. Ration. Mech. Anal., 163 (2002), 259-293. doi: 10.1007/s002050200201. Google Scholar [2] 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 [3] 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 [4] S. Bae, Positive entire solutions of semilinear elliptic equations with quadratically vanishing coefficient, J. Differential Equations, 237 (2007), 159-197. doi: 10.1016/j.jde.2007.03.003. Google Scholar [5] S. Bae, Entire solutions with asymptotic self-similarity for elliptic equations with exponential nonlinearity, J. Math. Anal. Appl., 428 (2015), 1085-1116. doi: 10.1016/j.jmaa.2015.03.036. Google Scholar [6] 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 [7] S. Bae and Y. Naito, Existence and separation of positive radial solutions for semilinear elliptic equations, J. Differential Equations, 257 (2014), 2430-2463. doi: 10.1016/j.jde.2014.05.042. Google Scholar [8] S. Bae and Y. Naito, Separation structure of radial solutions for semilinear elliptic equations with exponential nonlinearity, Discrete Contin. Dyn. Syst., 38 (2018), 4537-4554. doi: 10.3934/dcds.2018198. Google Scholar [9] C. Chen and H. Wang, Liouville theorems for the weighted Lane-Emden equation with finite Morse indices, Math. Methods Appl. Sci., 40 (2017), 4674-4682. Google Scholar [10] E. N. Dancer, Y. Du and Z. Guo, Finite Morse index solutions of an elliptic equation with supercritical exponent, J. Differential Equations, 250 (2011), 3281-3310. doi: 10.1016/j.jde.2011.02.005. Google Scholar [11] Y. Deng, Y. Li and F. Yang, On the positive radial solutions of a class of singular semilinear elliptic equations, J. Differential Equations, 253 (2012), 481-501. doi: 10.1016/j.jde.2012.02.017. Google Scholar [12] L. Dupaigne and A. Farina, Stable solutions of $-\Delta u = f(u)$ in $\mathbb{R}^{N}$, J. Eur. Math. Soc. (JEMS), 12 (2010), 855-882. doi: 10.4171/JEMS/217. Google Scholar [13] A. Farina, On the classification of solutions of the Lane-Emden equation on unbounded domains of $\mathbb{R}^{N}$, J. Math. Pures Appl. (9), 87 (2007), 537-561. doi: 10.1016/j.matpur.2007.03.001. Google Scholar [14] A. Farina, Stable solutions of $-\Delta u = e^{u}$ on $\mathbb{R}^{N}$, C. R. Math. Acad. Sci. Paris, 345 (2007), 63-66. doi: 10.1016/j.crma.2007.05.021. Google Scholar [15] A. Farina, Some symmetry results and Liouville-type theorems for solutions to semilinear equations, Nonlinear Anal., 121 (2015), 223-229. doi: 10.1016/j.na.2015.02.004. Google Scholar [16] A. Farina, B. Sciunzi and E. Valdinoci, Bernstein and De Giorgi type problems: New results via a geometric approach, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 7 (2008), 741-791. Google Scholar [17] C. Gui, Positive entire solutions of the equation $\Delta u+f(x,u) = 0$, J. Differential Equations, 99 (1992), 245-280. doi: 10.1016/0022-0396(92)90023-G. Google Scholar [18] C. Gui, K.-M. Ni and X. Wang, On the stability and instability of positive steady states of a semilinear heat equation in $\mathbb{R}^{n}$, Comm. Pure Appl. Math., 45 (1992), 1153-1181. doi: 10.1002/cpa.3160450906. Google Scholar [19] C. Gui, K.-M. Ni and X. Wang, Further study on a nonlinear heat equation, J. Differential Equations, 169 (2001), 588-613. doi: 10.1006/jdeq.2000.3909. Google Scholar [20] H. Hajlaoui, A. Harrabi and F. Mtiri, Liouville theorems for stable solutions of the weighted Lane-Emden system, Discrete Contin. Dyn. Syst., 37 (2017), 265-279. doi: 10.3934/dcds.2017011. Google Scholar [21] D. D. Joseph and T. S. Lundgren, Quasilinear Dirichlet problems driven by positive sources, Arch. Rational Mech. Anal., 49 (1972/73), 241-269. doi: 10.1007/BF00250508. Google Scholar [22] 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 [23] Y. Li, On the positive solutions of the Matukuma equation, Duke Math. J., 70 (1993), 575-589. doi: 10.1215/S0012-7094-93-07012-3. Google Scholar [24] Y. Liu, Y. Li and Y. Deng, Separation property of solutions for a semilinear elliptic equation, J. Differential Equations, 163 (2000), 381-406. doi: 10.1006/jdeq.1999.3735. Google Scholar [25] Y. Miyamoto, Intersection properties of radial solutions and global bifurcation diagrams for supercritical quasilinear elliptic equations, NoDEA Nonlinear Differential Equations Appl., 23 (2016), Art. 16, 24 pp. doi: 10.1007/s00030-016-0359-0. Google Scholar [26] Y. Miyamoto and K. Takahashi, Generalized Joseph-Lundgren exponent and intersection properties for supercritical quasilinear elliptic equations, Arch. Math. (Basel), 108 (2017), 71-83. doi: 10.1007/s00013-016-0969-0. Google Scholar [27] W.-M. Ni and S. Yotsutani, On Matukuma's equation and related topics, Proc. Japan Acad. Ser. A Math. Sci., 62 (1986), 260-263. doi: 10.3792/pjaa.62.260. Google Scholar [28] 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 [29] J. I. Tello, Stability of steady states of the Cauchy problem for the exponential reaction-diffusion equation, J. Math. Anal. Appl., 324 (2006), 381-396. doi: 10.1016/j.jmaa.2005.12.011. Google Scholar [30] S. Villegas, Asymptotic behavior of stable radial solutions of semilinear elliptic equations in $\mathbb{R}^{N}$, J. Math. Pures Appl. (9), 88 (2007), 241-250. doi: 10.1016/j.matpur.2007.06.004. Google Scholar [31] X. Wang, On the Cauchy problem for reaction-diffusion equations, Trans. Amer. Math. Soc., 337 (1993), 549-590. doi: 10.1090/S0002-9947-1993-1153016-5. Google Scholar [32] E. Yanagida, Structure of radial solutions to $\Delta u+K(|x|)|u|^{p-1}u=0$ in $\mathbb{R}^{N}$, SIAM J. Math. Anal., 27 (1996), 997-1014. doi: 10.1137/0527053. Google Scholar
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