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

• Fulltext

https://www.ias.ac.in/article/fulltext/pmsc/123/01/0145-0150

• Keywords

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

• Abstract

In this work, we consider the positive solutions to the singular problem

where $0 &lt; \alpha &lt; 1,a&gt;0$ and $c&gt;0$ are constants, 𝛺 is a bounded domain with smooth boundary $\partial\Omega,\Delta$ is a Laplacian operator, and $f:[0,\infty]\longrightarrow\mathbb{R}$ is a continuous function. The weight functions $m(x)$ satisfies $m(x)\in C(\Omega)$ and $m(x)&gt;m_0&gt;0$ for $x\in\Omega$ and also $\|m\|_\infty=l &lt; \infty$. We assume that there exist $A&gt;0, M&gt;0,p&gt;1$ such that $alu-M\leq f(u)\leq Au^p$ for all $u\in[0,\infty)$. We prove the existence of a positive solution via the method of sub-supersolutions when $m_0 a&gt;\frac{2\lambda_1}{1+\alpha}$ and 𝑐 is small. Here $\lambda_1$ is the first eigenvalue of operator $-\Delta$ with Dirichlet boundary conditions.

• Author Affiliations

1. Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran
2. Department of Mathematics and Informatics, Quang Binh University, 312 Ly Thuong Kiet, Dong Hoi, Quang Binh, Vietnam

• Proceedings – Mathematical Sciences

Volume 132, 2022
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Posted on July 25, 2019