EXISTENCE OF MULTIPLE POSITIVE SOLUTIONS FOR A SCHRO ̈DINGER-TYPE SINGULAR FALLING ZERO PROBLEM
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Extending \cite{K23}, we establish the existence of multiple positive solutions for a Schr\"odinger-type singular elliptic equation:$$\left\{\begin{array}{rlll}-\Delta u +V(x) u&=&\lambda \frac{ f(u)}{u^\beta}, & x \in\Omega,\\u &=&0, & x \in \partial\Omega,\end{array}\right. $$where $0 \in \Omega$ is a bounded domain in $ \mathbb{R}^N, N\geq 1,$ with a smooth boundary $\partial\Omega,$ $\beta \in [0,1),$$f \in C[0,\infty)$, $V:\Omega \rightarrow \mathbb{R} $ is a bounded function and$\lambda$ is a positive parameter. In particular, when $f (s) >0 $ on $[0, \sigma) $ and $f(s) <0$ for $s > \sigma,$ we establish the existence of at least three positive solutions for a certain range of $\lambda$ byusing the method of sub and supersolutions.
Abstract
Extending \cite{K23}, we establish the existence of multiple positive solutions for a Schr\"odinger-type singular elliptic equation:$$\left\{\begin{array}{rlll}-\Delta u +V(x) u&=&\lambda \frac{ f(u)}{u^\beta}, & x \in\Omega,\\u &=&0, & x \in \partial\Omega,\end{array}\right. $$where $0 \in \Omega$ is a bounded domain in $ \mathbb{R}^N, N\geq 1,$ with a smooth boundary $\partial\Omega,$ $\beta \in [0,1),$$f \in C[0,\infty)$, $V:\Omega \rightarrow \mathbb{R} $ is a bounded function and$\lambda$ is a positive parameter. In particular, when $f (s) >0 $ on $[0, \sigma) $ and $f(s) <0$ for $s > \sigma,$ we establish the existence of at least three positive solutions for a certain range of $\lambda$ byusing the method of sub and supersolutions.
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