带有退化强制项的非线性椭圆方程解的存在性与不存在性  

Existence and Nonexistence of Solutions for Nonlinear Elliptic Equations with Degenerate Coercivity

作　　者：米清清 严畅 钮维生[1] MI Qingqing;YAN Chang;NIU Weisheng(School of Mathematical Sciences,Anhui University,Hefei 230601,China)

机构地区：[1]安徽大学数学科学学院,安徽合肥230601

出　　处：《应用数学》2022年第4期835-846,共12页Mathematica Applicata

基　　金：国家自然科学基金资助项目(11701002)。

摘　　要：本文研究带有退化强制主部算子的非线性椭圆型方程解的存在性与不存在性.对于外力项为可积函数的情形,首先对方程进行适当的光滑逼近并得到逼近方程有界弱解的存在性,随后通过选取适当的试验函数,结合非线性项的增长次数建立逼近解的一致正则性估计,进而利用适当的紧性定理给出逼近解序列的收敛性,从而得到方程有限能量弱解的存在性;对于外力项为奇异Radon测度的情形,我们结合适当的测度分解技巧与逼近解的收敛性证明熵解的不存在性.We investigate the existence and nonexistence of solutions to nonlinear elliptic equations with degenerate coercivity.Our results are twofold.When the external force term is integrable,we first perform appropriate smooth approximation and obtain the existence of the bounded weak solution of the approximate equation.Moreover,by choosing proper test functions and taking advantage of orders of the nonlinear term,we establish some uniform regularity estimates for the approximate solution.Then by using proper compactness theorem,we derive the convergence of the approximate solution sequence,and finally the existence of finite energy weak solutions for the original equation.When the external force term is a singular Radon measure,we prove the nonexistence of the entropy solution by combining some measure decomposition techniques with the convergence of the approximate solution.

关 键 词：退化椭圆方程 弱解 存在性 熵解 不存在性 

分 类 号：O175.25[理学—数学]