• N T Chung

Articles written in Proceedings – Mathematical Sciences

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

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

\begin{equation*}\begin{cases}-\Delta u=am(x)u-f(u)-\frac{c}{u^\alpha} &amp; \text{in}\quad\Omega,\\ u=0 &amp; \text{on}\quad\partial\Omega,\end{cases}\end{equation*}

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.

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