一类非线性双调和方程在${{\mathbb{R}}^{N}}$上正整解存在的充分必要条件

Abstract

Abstract:

The aim of this paper is to study the nonlinear biharmonic equations of the following form $\triangle^2u=f(|x|, u, |\nabla u|)(x\in \mathbb{R} ^N, N>2)$. The Sufficient and necessary condition for the existence of positive entire solutions is proved, and some properties of the solutions are obtained.

Key words: Nonlinear biharmonic equation, Positive entire solution, Close convex subset, Equicontinuity, Fixed point theorem

CLC Number:

O175.25

Xiaohang Ou. Sufficient and Necessary Condition for the Existence of Positive Entire Solutions of a Nonlinear Biharmonic Equations on ${{\mathbb{R}}^{N}}$[J].Acta mathematica scientia,Series A, 2019, 39(3): 535-544.

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References 9

1	Kusano T , Swanson C A . Positive entire solutions of semilinear biharmonic equations. Hiroshima Math J, 1987, 17: 13- 28 doi: 10.32917/hmj/1206130185
2	文如庆, 陈世明. 一类半线性双调和方程的正整解. 应用数学, 1994, 7 (1): 85- 92
	Wen R Q , Chen S M . The positive entire solutions for a class of semlinear biharmonic equations. Mathematica Applica, 1994, 7 (1): 85- 92
3	许兴业. 关于${{\mathbb{R}}^{N}}$中奇异非线性双调和方程正整解的存在性. 南京大学学报数学半年刊, 1998, 15 (1): 99- 105
	Xu X Y . The existence of positive entire solutions about singular nonlinear biharmonic equations on ${{\mathbb{R}}^{N}}$. Journal of Nanjing University Mathematical Biquarterly, 1998, 15 (1): 99- 105
4	许兴业. ${{\mathbb{R}}^{N}}$上带奇异性的非线性双调和方程的正整解. 数学物理学报, 2009, 29 (1): 87- 93
	Xu X Y . The positive entire solutions for a class of nonlinear biharmonic equations with singular on ${{\mathbb{R}}^{N}}$. Acta Mathematica Scientia, 2009, 29 (1): 87- 93
5	叶长青. 一类${{\mathbb{R}}^{N}}$(N ≥ 3)上奇异非线性双调和方程正整解. 系统科学与数学, 2007, 27 (6): 892- 898 doi: 10.3969/j.issn.1000-0577.2007.06.010
	Ye C Q . Positive entire solutions for a class of singular nonlinear biharmonic equations on ${{\mathbb{R}}^{N}}$(N ≥ 3). Journal of System Science and Mathematical Sciences, 2007, 27 (6): 892- 898 doi: 10.3969/j.issn.1000-0577.2007.06.010
6	叶长青. 奇异非线性双调和方程存在正整解的充分必要条件. 闽南师范大学学报(自然科学版), 2016, (1): 6- 13 doi: 10.3969/j.issn.1008-7826.2016.01.002
	Ye C Q . Existence and necessary for the existence of positive entire solutions for a class singular nonlinear biharmonic equations. Journal of Minnan University (Nat Sci), 2016, (1): 6- 13 doi: 10.3969/j.issn.1008-7826.2016.01.002
7	欧笑杭. 关于非线性方程$div(\|Du\|^{p-2}Du)=f(\|x\|, u, \|\nabla u\|)$的正整解. 广东第二师范学院学报, 2017, 37 (3): 54- 58 doi: 10.3969/j.issn.2095-3798.2017.03.007
	Ou X H . The positiver entire solutions to the nonlinear equations such as $div(\|Du\|^{p-2}Du)=f(\|x\|, u, \|\nabla u\|)$. Journal of Guangdong University of Education, 2017, 37 (3): 54- 58 doi: 10.3969/j.issn.2095-3798.2017.03.007
8	Adams R A . Sobolev Spaces. New York: Academic Press, 1975
9	Edwards R E . Functional Analysis. New York: Holt Rinehart and Winston, 1965

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Sufficient and Necessary Condition for the Existence of Positive Entire Solutions of a Nonlinear Biharmonic Equations on ${{\mathbb{R}}^{N}}$

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