scholarly journals Radial symmetry of entire solutions of a bi-harmonic equation with exponential nonlinearity

2015 ◽  
Vol 268 (7) ◽  
pp. 1972-2004 ◽  
Author(s):  
Zongming Guo ◽  
Xia Huang ◽  
Feng Zhou
Keyword(s):  
Radial Symmetry ◽  
Entire Solutions ◽  
Exponential Nonlinearity ◽  
Harmonic Equation

Radial symmetry of non-maximal entire solutions of a bi-harmonic equation with exponential nonlinearity

2019 ◽  
Vol 149 (6) ◽  
pp. 1603-1625
Author(s):  
Hongxia Guo ◽  
Zongming Guo ◽  
Fangshu Wan
Keyword(s):  
Sufficient Conditions ◽  
Entire Solution ◽  
Radial Symmetry ◽  
Necessary And Sufficient Conditions ◽  
Entire Solutions ◽  
Exponential Nonlinearity ◽  
Image Position ◽  
Harmonic Equation ◽  
Necessary And Sufficient

AbstractWe study radial symmetry of entire solutions of the equation0.1$$\Delta ^2u = 8(N-2)(N-4)e^u\quad {\rm in}\;R^N\;\;(N \ges 5).$$It is known that (0.1) admits infinitely many radially symmetric entire solutions. These solutions may have either a (negative) logarithmic behaviour or a (negative) quadratic behaviour at infinity. Up to translations, we know that there is only one radial entire solution with the former behaviour, which is called ‘maximal radial entire solution’, and infinitely many radial entire solutions with the latter behaviour, which are called ‘non-maximal radial entire solutions’. The necessary and sufficient conditions for an entire solutionuof (0.1) to be the maximal radial entire solution are presented in [7] recently. In this paper, we will give the necessary and sufficient conditions for an entire solutionuof (0.1) to be a non-maximal radial entire solution.

Radial Symmetry of Entire Solutions of a Biharmonic Equation with Supercritical Exponent

2019 ◽  
Vol 19 (2) ◽  
pp. 291-316
Author(s):  
Zongming Guo ◽  
Long Wei
Keyword(s):  
Asymptotic Behavior ◽  
Biharmonic Equation ◽  
Sufficient Conditions ◽  
Entire Solution ◽  
Radial Symmetry ◽  
Entire Solutions ◽  
Moving Plane ◽  
Exact Asymptotic Behavior ◽  
Necessary And Sufficient ◽  
Positive Entire Solution

AbstractNecessary and sufficient conditions for a regular positive entire solution u of a biharmonic equation\Delta^{2}u=u^{p}\quad\text{in }\mathbb{R}^{N},\,N\geq 5,\,p>\frac{N+4}{N-4}to be a radially symmetric solution are obtained via the exact asymptotic behavior of u at {\infty} and the moving plane method (MPM). It is known that above equation admits a unique positive radial entire solution {u(x)=u(|x|)} for any given {u(0)>0}, and the asymptotic behavior of {u(|x|)} at {\infty} is also known. We will see that the behavior similar to that of a radial entire solution of above equation at {\infty}, in turn, determines the radial symmetry of a general positive entire solution {u(x)} of the equation. To make the procedure of the MPM work, the precise asymptotic behavior of u at {\infty} is obtained.

Infinite multiplicity of stable entire solutions for a semilinear elliptic equation with exponential nonlinearity

2019 ◽  
Vol 149 (5) ◽  
pp. 1371-1404
Author(s):  
Soohyun Bae
Keyword(s):  
Elliptic Equation ◽  
Semilinear Elliptic Equation ◽  
Strong Maximum Principle ◽  
Entire Solutions ◽  
Barrier Method ◽  
Exponential Nonlinearity ◽  
Infinite Multiplicity ◽  
The Stability ◽  
Linearized Operator ◽  
Asymptotic Expansion Of Solutions

AbstractWe consider the infinite multiplicity of entire solutions for the elliptic equation Δu + K(x)eu + μf(x) = 0 in ℝn, n ⩾ 3. Under suitable conditions on K and f, the equation with small μ ⩾ 0 possesses a continuum of entire solutions with a specific asymptotic behaviour. Typically, K behaves like |x|ℓ at ∞ for some ℓ > −2 and the entire solutions behave asymptotically like − (2 + ℓ)log |x| near ∞. Main tools of the analysis are comparison principle for separation structure, asymptotic expansion of solutions near ∞, barrier method and strong maximum principle. The linearized operator for the equation has two characteristic behaviours related with the stability and the weak asymptotic stability of the solutions as steady states for the corresponding parabolic equation.

Nonexistence Results of Positive Entire Solutions for Quasilinear Elliptic Inequalities

1997 ◽  
Vol 40 (2) ◽  
pp. 244-253 ◽  
Author(s):  
Yūki Naito ◽  
Hiroyuki Usami
Keyword(s):  
Sufficient Conditions ◽  
Radial Symmetry ◽  
Entire Solutions ◽  
Elliptic Inequality ◽  
Quasilinear Elliptic ◽  
Elliptic Inequalities

AbstractThis paper treats the quasilinear elliptic inequalitywhere N ≥ 2, m > 1, σ >m− 1, and p:ℝN → (0, ∞) is continuous. Sufficient conditions are given for this inequality to have no positive entire solutions. When p has radial symmetry, the existence of positive entire solutions can be characterized by our results and some known results.

On the Radial Symmetry Property for Harmonic Functions

2020 ◽  
Vol 64 (10) ◽  
pp. 9-19
Author(s):  
V. V. Volchkov ◽  
Vit. V. Volchkov
Keyword(s):  
Harmonic Functions ◽  
Symmetry Property ◽  
Radial Symmetry

Pollen characterization of ornamental species of Mammillaria Haw. (Cactaceae/Cactoideae)

2019 ◽  
Vol 6 (1) ◽  
pp. 13 ◽  
Author(s):  
Denise M. D. S. Mouga ◽  
Gabriel R. Schroeder ◽  
Nilton P. Vieira Junior ◽  
Enderlei Dec
Keyword(s):  
Pollen Morphology ◽  
Pollen Grains ◽  
Radial Symmetry ◽  
Morphological Data ◽  
Medium Size ◽  
Ornamental Species

The pollen morphology of thirteen species of Cactaceae was studied: M. backebergiana F.G. Buchenau, M. decipiens Scheidw, M. elongata DC, M. gracilis Pfeiff., M. hahniana Werderm., M. marksiana Krainz, M. matudae Bravo, M. nejapensis R.T. Craig & E.Y. Dawson, M. nivosa Link ex Pfeiff., M. plumosa F.A.C. Weber, M. prolifera (Mill.) Haw, M. spinosissima var. “A Peak” Lem. and M. voburnensis Scheer. All analysed pollen grains are monads, with radial symmetry, medium size (M. gracilis, M. marksiana, M. prolifera, large), tricolpates (dimorphs in M. plumosa [3-6 colpus] and M. prolifera [3-6 colpus]), with circular-subcircular amb (quadrangular in M. prolifera and M. plumosa with six colpus). The pollen grains presented differences in relation to the shape and exine thickness. The exine was microechinate and microperforated. The pollen morphological data are unpublished and will aid in studies that use pollen samples. These pollen grains indicate ornamental cacti.

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