This paper deals with the existence of positive solutions to the singular elliptic boundary value problem involving p-Laplace operator
−div(|∇u|p−2∇u)=h(x)uα+k(x)uβ,x∈Ω;u(x)>0,x∈Ω;u(x)=0,x∈∂Ω;\documentclass[12pt]{minimal}
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\begin{document}$$-\operatorname{div}\bigl(|\nabla u|^{p-2}\nabla u\bigr)= \frac{h(x)}{u^{\alpha}}+k(x)u^{\beta},\ x\in\Omega;\ \ u(x)>0,\ x\in\Omega;\ \ u(x)=0,\ x \in \partial\Omega; $$\end{document} where Ω⊂RN\documentclass[12pt]{minimal}
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\begin{document}$\Omega\subset\mathbb{R}^{N}$\end{document} (N≥1\documentclass[12pt]{minimal}
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\begin{document}$N\geq1$\end{document}) is a bounded domain with smooth boundary ∂Ω, h∈L1(Ω)\documentclass[12pt]{minimal}
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\begin{document}$h\in L^{1}(\Omega)$\end{document}, h(x)>0\documentclass[12pt]{minimal}
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\begin{document}$h(x)>0$\end{document} almost everywhere in Ω, k∈L∞(Ω)\documentclass[12pt]{minimal}
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\begin{document}$k\in L^{\infty}(\Omega)$\end{document} is nonnegative, p>2\documentclass[12pt]{minimal}
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\begin{document}$p>2$\end{document}, α>1\documentclass[12pt]{minimal}
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\begin{document}$\alpha>1$\end{document} and β∈(0,p−1)\documentclass[12pt]{minimal}
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\begin{document}$\beta\in(0,p-1)$\end{document}. A compatibility condition on the couple (h(x),α)\documentclass[12pt]{minimal}
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\begin{document}$(h(x),\alpha)$\end{document} is given for the problem to have at least one solution. More precisely, it is shown that the problem admits a solution if and only if there exists u0∈H01(Ω)\documentclass[12pt]{minimal}
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\begin{document}$u_{0}\in H_{0}^{1}(\Omega)$\end{document} such that ∫Ωhu01−αdx<∞\documentclass[12pt]{minimal}
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\begin{document}$\int_{\Omega}hu_{0}^{1-\alpha}\,\mathrm{d}x<\infty$\end{document}.
