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September  2018, 38(9): 4537-4554. doi: 10.3934/dcds.2018198

## Separation structure of radial solutions for semilinear elliptic equations with exponential nonlinearity

 1 Department of Mathematical Sciences, Hanbat National University, Daejeon 34158, Republic of Korea 2 Department of Mathematics, Ehime University, Matsuyama 790-8577, Japan

* Corresponding author: Y. Naito

Received  November 2017 Revised  April 2018 Published  June 2018

We consider the semilinear elliptic equation $Δ u + K(|x|)e^u = 0$ in $\mathbf{R}^N$ for $N > 2$, and investigate separation phenomena of radial solutions. In terms of intersection and separation, we classify the solution structures and establish characterizations of the structures. These observations lead to sufficient conditions for partial separation. For $N = 10+4\ell$ with $\ell>-2$, the equation changes its nature drastically according to the sign of the derivative of $r^{-\ell}K(r)$ when $r^{-\ell}K(r)$ is monotonic in $r$ and $r^{-\ell} K(r)\to1$ as $r\to∞$.

Citation: Soohyun Bae, Yūki Naito. Separation structure of radial solutions for semilinear elliptic equations with exponential nonlinearity. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4537-4554. doi: 10.3934/dcds.2018198
##### References:
 [1] S. Bae, Separation structure of positive radial solutions of a semilinear elliptic equation in $\mathbf{R}^n$, J. Differential Equations, 194 (2003), 460-499.  doi: 10.1016/S0022-0396(03)00172-4.  Google Scholar [2] S. Bae, Infinite multiplicity and separation structure of positive solutions for a semilinear elliptic equation in $\mathbf{R}^N$, J. Differential Equations, 200 (2004), 274-311.  doi: 10.1016/j.jde.2003.11.006.  Google Scholar [3] S. Bae, Infinite multiplicity of stable entire solutions for a semilinear elliptic equation with exponential nonlinearity, to appear in Proc. Roy. Soc. Edinburgh Sect. A. Google Scholar [4] 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 [5] S. Bae and T.-K. Chang, On a class of semilinear elliptic equations in $\mathbf{R}^N$, J. Differential Equations, 185 (2002), 225-250.  doi: 10.1006/jdeq.2001.4162.  Google Scholar [6] 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 [7] K.-S. Cheng and J.-T. Lin, On the elliptic equations $Δ u = K(x)u^{σ}$ and $Δ u = K(x)e^{2u}$, Trans. Amer. Math. Soc., 304 (1987), 639-668.  doi: 10.1090/S0002-9947-1987-0911088-1.  Google Scholar [8] K.-S. Cheng and W.-M. Ni, On the structure of the conformal Gaussian curvature equation on $\mathbf{R}^2$, Duke Math. J., 62 (1991), 721-737.  doi: 10.1215/S0012-7094-91-06231-9.  Google Scholar [9] W.-Y. Ding and W.-M. Ni, On the elliptic equation $Δ u + K u^{(n+2)/(n-2)} =0$ and related topics, Duke Math. J., 52 (1985), 485-506.  doi: 10.1215/S0012-7094-85-05224-X.  Google Scholar [10] C. Gui, W.-M. Ni and X. Wang, On the stability and instability of positive steady states of a semilinear heat equation in $\mathbf{R}^N$, Comm. Pure Appl. Math., 45 (1992), 1153-1181.  doi: 10.1002/cpa.3160450906.  Google Scholar [11] C. Gui, W.-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 [12] D. D. Joseph and T. S. Lundgren, Quasilinear Dirichlet problems driven by positive sources, Arch. Rational Mech. Anal., 49 (1973), 241-269.  doi: 10.1007/BF00250508.  Google Scholar [13] Y. Li and W.-M. Ni, On the existence and symmetry properties of finite total mass solutions of the Matukuma equation, the Eddington equation and their generalizations, Arch. Rational Mech. Anal., 108 (1989), 175-194.  doi: 10.1007/BF01053462.  Google Scholar [14] 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 [15] W.-M. Ni, On the elliptic equation $Δ u + K(x)e^{2u} = 0$ and conformal metrics with prescribed Gaussian curvatures, Invent. Math., 66 (1982), 343-352.  doi: 10.1007/BF01389399.  Google Scholar [16] 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 [17] 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 [18] 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

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##### References:
 [1] S. Bae, Separation structure of positive radial solutions of a semilinear elliptic equation in $\mathbf{R}^n$, J. Differential Equations, 194 (2003), 460-499.  doi: 10.1016/S0022-0396(03)00172-4.  Google Scholar [2] S. Bae, Infinite multiplicity and separation structure of positive solutions for a semilinear elliptic equation in $\mathbf{R}^N$, J. Differential Equations, 200 (2004), 274-311.  doi: 10.1016/j.jde.2003.11.006.  Google Scholar [3] S. Bae, Infinite multiplicity of stable entire solutions for a semilinear elliptic equation with exponential nonlinearity, to appear in Proc. Roy. Soc. Edinburgh Sect. A. Google Scholar [4] 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 [5] S. Bae and T.-K. Chang, On a class of semilinear elliptic equations in $\mathbf{R}^N$, J. Differential Equations, 185 (2002), 225-250.  doi: 10.1006/jdeq.2001.4162.  Google Scholar [6] 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 [7] K.-S. Cheng and J.-T. Lin, On the elliptic equations $Δ u = K(x)u^{σ}$ and $Δ u = K(x)e^{2u}$, Trans. Amer. Math. Soc., 304 (1987), 639-668.  doi: 10.1090/S0002-9947-1987-0911088-1.  Google Scholar [8] K.-S. Cheng and W.-M. Ni, On the structure of the conformal Gaussian curvature equation on $\mathbf{R}^2$, Duke Math. J., 62 (1991), 721-737.  doi: 10.1215/S0012-7094-91-06231-9.  Google Scholar [9] W.-Y. Ding and W.-M. Ni, On the elliptic equation $Δ u + K u^{(n+2)/(n-2)} =0$ and related topics, Duke Math. J., 52 (1985), 485-506.  doi: 10.1215/S0012-7094-85-05224-X.  Google Scholar [10] C. Gui, W.-M. Ni and X. Wang, On the stability and instability of positive steady states of a semilinear heat equation in $\mathbf{R}^N$, Comm. Pure Appl. Math., 45 (1992), 1153-1181.  doi: 10.1002/cpa.3160450906.  Google Scholar [11] C. Gui, W.-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 [12] D. D. Joseph and T. S. Lundgren, Quasilinear Dirichlet problems driven by positive sources, Arch. Rational Mech. Anal., 49 (1973), 241-269.  doi: 10.1007/BF00250508.  Google Scholar [13] Y. Li and W.-M. Ni, On the existence and symmetry properties of finite total mass solutions of the Matukuma equation, the Eddington equation and their generalizations, Arch. Rational Mech. Anal., 108 (1989), 175-194.  doi: 10.1007/BF01053462.  Google Scholar [14] 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 [15] W.-M. Ni, On the elliptic equation $Δ u + K(x)e^{2u} = 0$ and conformal metrics with prescribed Gaussian curvatures, Invent. Math., 66 (1982), 343-352.  doi: 10.1007/BF01389399.  Google Scholar [16] 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 [17] 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 [18] 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
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