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We prove the existence of a positive solution $u_{q}$ for the problem $(P_{a,q})$:\n  $-\\Delta u=a(x)u^{q}$ in $\\Omega$, $\\frac{\\partial u}{\\partial\\nu}=0$ on $\\partial\\Omega$,\n  if $q_{0}<q<1$, for some $q_{0}=q_{0}(a)>0$. In doing so, we improve the existence result previously established in [16]. In addition, we provide the asymptotic behavior of $u_{q}$ as $q\\rightarrow1^{-}$. 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We prove the existence of a positive solution $u_{q}$ for the problem $(P_{a,q})$:\n  $-\\Delta u=a(x)u^{q}$ in $\\Omega$, $\\frac{\\partial u}{\\partial\\nu}=0$ on $\\partial\\Omega$,\n  if $q_{0}<q<1$, for some $q_{0}=q_{0}(a)>0$. In doing so, we improve the existence result previously established in [16]. In addition, we provide the asymptotic behavior of $u_{q}$ as $q\\rightarrow1^{-}$. 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