Remark on an infinite semipositone problem with indefinite weight and falling zeros |
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Authors: | G A AFROUZI S SHAKERI N T CHUNG |
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Institution: | 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
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Abstract: | In this work, we consider the positive solutions to the singular problem $$ \left\{\begin{array}{ll} -\Delta u = am(x)u-f(u) - \dfrac{c}{u^{\alpha}} & {\rm in}\Omega,\\ u=0 & {\rm on} \partial\Omega, \end{array} \right. $$ where 0?<?α?<?1,a?>?0 and c?>?0 are constants, Ω is a bounded domain with smooth boundary $\partial\Omega$ , Δ is a Laplacian operator, and $f:0,\infty] \longrightarrow{\mathbb R}$ is a continuous function. The weight functions m(x) satisfies m(x)?∈?C(Ω) and m(x)?>?m 0?>?0 for x?∈?Ω and also ||m||?∞??=?l?<?∞. We assume that there exist A?>?0, M?>?0, p?>?1 such that alu???M?≤?f(u)?≤?Au p for all u?∈?0,?∞?). We prove the existence of a positive solution via the method of sub-supersolutions when $m_{0}a>\frac{2\lambda_{1} }{1+\alpha}$ and c is small. Here λ 1 is the first eigenvalue of operator ??Δ with Dirichlet boundary conditions. |
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