• East Asian mathematical journal
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• v.36 no.5
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• pp.605-612
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• 2020

We establish the existence of a positive solution to a semipositone system with integral boundary condition for the large value of the parameter involved in the system. We prove our results by using sub and super solution argument.

• East Asian mathematical journal
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• v.40 no.3
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• pp.319-328
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• 2024

We establish an existence result for a positive solution to the Schrödinger-type singular semipositone problem: $-{\Delta}u\,=\,V(x)u\,=\,{\lambda}{\frac{f(u)}{u^{\alpha}}}$ in Ω, u = 0 on ∂Ω, where Ω is a bounded domain in ℝ^N , N > 2, λ ∈ ℝ is a positive parameter, V ∈ L^∞(Ω), 0 < α < 1, f ∈ C([0, ∞), ℝ) with f(0) < 0. In particular, when ${\frac{f(s)}{s^{\alpha}}}$ is sublinear at infinity, we establish the existence of a positive solutions for λ ≫ 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.

• Journal of applied mathematics & informatics
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• v.28 no.1_2
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• pp.163-173
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• 2010

We consider the system $$\{{{-{\Delta}_pu\;=\;{\lambda}f(\upsilon),\;\;\;x\;{\in}\;{\Omega}, \atop -{\Delta}_q{\upsilon}\;=\;{\mu}g(u),\;\;\;x\;{\in}\;{\Omega},} \atop u\;=\;\upsilon\;=\;0,\;\;\;x\;{\in}\;{\partial\Omega},}$$ where ${\Delta}_pu\;=\;div(|{\nabla}_u|^{p-2}{\nabla}_u)$, ${\Delta}_{q{\upsilon}}\;=\;div(|{\nabla}_{\upsilon}|^{q-2}{\nabla}_{\upsilon})$, p, $q\;{\geq}\;2$, $\Omega$ is a ball in $\mathbf{R}^N$ with a smooth boundary $\partial\Omega$, $N\;{\geq}\;1$, $\lambda$, $\mu$ are positive parameters, and f, g are smooth functions that are negative at the origin and f(x) ~ $x^m$ g(x) ~ $x^n$ for x large for some m, $n\;{\geq}\;0$ with mn < (p - 1)(q - 1). We establish the existence and uniqueness of positive radial solutions when the parameters $\lambda$ and $\mu$ are large.