自然增长条件下拟线性椭圆型方程解的存在性

李周欣;沈尧天;姚仰新;

数学学报 ›› 2009 ›› Issue (04) : 163-176.

数学学报 ›› 2009 ›› Issue (04) : 163-176. DOI: 10.12386/A2009sxxb0095
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自然增长条件下拟线性椭圆型方程解的存在性

    李周欣;沈尧天;姚仰新;
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Nontrivial Solutions for Quasilinear Elliptic Equations with Natural Growth

    Zhou Xin LI Yao Tian SHEN Yang Xin YAO Department of Mathematics,South China University of Technology,Guangzhou 510640,P.R.China E-mail:lizhouxin@tom.com;maytshen@scut.edu.cn;mayxyao@scut.edu.cn
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摘要

考虑自然增长条件下一类拟线性椭圆型方程的非平凡解的存在性和不存在性.在以往的工作中,对此类方程的一个重要假设是自然增长项和未知函数的乘积是非负的,本文改进了此条件,把此条件推广到小于零的情况,得到使得方程解存在时自然增长项和未知函数的乘积的一个下界,并利用不光滑泛函的临界点定理证明了非平凡解的存在性.进一步,利用Pohozaev恒等式证明了当自然增长项和未知函数的乘积小于此下界时,方程不存在解,从而说明了此下界是最佳的.

Abstract

We consider the existence and nonexistence of nontrivial solution for quasilinear elliptic equation under natural growth condition.In the previous works on such equation,one of the important assumptions is that the product of the natural growth term and the unknown function is not less than zero.We improve this condition,extend it to the minus part and get the boundary below which such equation has nontrivial solution.We prove the existence of the nontrivial solution via nonsmooth critical point theory.Moreover,we use the Pohozaev identity to prove that the equation does not have any solution when the product of the natural growth term and the unknown function is less than this boundary,which means that this boundary is the best one.

关键词

椭圆型方程 / 不光滑临界点定理 / Pohozaev恒等式 / 自然增长条件

Key words

nonsmooth critical point theory / Pohozaev identity / elliptic equation / natural growth condition

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李周欣;沈尧天;姚仰新;. 自然增长条件下拟线性椭圆型方程解的存在性. 数学学报, 2009(04): 163-176 https://doi.org/10.12386/A2009sxxb0095
Zhou Xin LI Yao Tian SHEN Yang Xin YAO Department of Mathematics,South China University of Technology,Guangzhou 510640,P.R.China E-mail:lizhouxin@tom.com;maytshen@scut.edu.cn;mayxyao@scut.edu.cn. Nontrivial Solutions for Quasilinear Elliptic Equations with Natural Growth. Acta Mathematica Sinica, Chinese Series, 2009(04): 163-176 https://doi.org/10.12386/A2009sxxb0095

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