Remark on an Infinite Semipositone Problem with Indefinite Weight and Falling Zeros

Proceedings Mathematical Sciences

G A Afrouzi³  S Shakeri²  N T Chung¹ 

[1] $$;Department of Mathematics and Informatics, Quang Binh University, Ly Thuong Kiet, Dong Hoi, Quang Binh, Vietnam$$;Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran$$

关键词: Infinite semipositone problems;    indefinite weight;    falling zeros;    sub-supersolution method.;   

DOI  :  

学科分类：数学（综合）

来源: Indian Academy of Sciences

【 摘 要 】

In this work, we consider the positive solutions to the singular problemegin{equation*}egin{cases}-𝛥 u=am(x)u-f(u)-frac{c}{u^𝛼} & ext{in}quad𝛺, u=0 & ext{on}quad𝜕𝛺,end{cases}end{equation*}where $0 < 𝛼 < 1,a>0$ and $c>0$ are constants, 𝛺 is a bounded domain with smooth boundary 𝜕𝛺,𝛥 is a Laplacian operator, and $f:[0,∞]longrightarrowmathbb{R}$ is a continuous function. The weight functions $m(x)$ satisfies $m(x)in C(𝛺)$ and $m(x)>m_0>0$ for $xin𝛺$ and also $|m|_∞=l < ∞$. We assume that there exist $A>0, M>0,p>1$ such that $alu-M≤ f(u)≤ Au^p$ for all $uin[0,∞)$. We prove the existence of a positive solution via the method of sub-supersolutions when $m_0 a>frac{2𝜆_1}{1+𝛼}$ and 𝑐 is small. Here 𝜆1 is the first eigenvalue of operator -𝛥 with Dirichlet boundary conditions.

【 授权许可】

Unknown