Existence of a positive solution to infinite semipositone problems
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We establish an existence result for a positive solution to the Schr\"odinger-type singular semipositone problem:$-\Delta u+V(x)u=\lambda \frac{ f(u)}{u^\alpha}$ in $ \Omega, ~u=0 $ on $\partial \Omega,$where $\Omega$ is a bounded domain in $ \mathbb{R}^N, N >2, \lambda \in \mathbb{R}$ is a positive parameter,$V \in L^\infty(\Omega)$, $0< \alpha <1$, $ f \in C([0, \infty), \mathbb{R})$ with $f(0) <0$. In particular, when $\frac{f(s)}{s^\alpha}$ is sublinear at infinity,% and the case when f satisfies sublinear growth condition,we establish the existence of a positive solutions for $\lambda \gg 1$. The proofs are mainly based on the sub and supersolution method. Further, we extend our existence result to infinite semipositone problems with mixed boundary conditions.
Abstract
We establish an existence result for a positive solution to the Schr\"odinger-type singular semipositone problem:$-\Delta u+V(x)u=\lambda \frac{ f(u)}{u^\alpha}$ in $ \Omega, ~u=0 $ on $\partial \Omega,$where $\Omega$ is a bounded domain in $ \mathbb{R}^N, N >2, \lambda \in \mathbb{R}$ is a positive parameter,$V \in L^\infty(\Omega)$, $0< \alpha <1$, $ f \in C([0, \infty), \mathbb{R})$ with $f(0) <0$. In particular, when $\frac{f(s)}{s^\alpha}$ is sublinear at infinity,% and the case when f satisfies sublinear growth condition,we establish the existence of a positive solutions for $\lambda \gg 1$. The proofs are mainly based on the sub and supersolution method. Further, we extend our existence result to infinite semipositone problems with mixed boundary conditions.
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