Abstract This paper deals with the problem { - Δ u = λ u q ( x ) , x ∈ Ω , u = 0 , x ∈ ∂ Ω , \left\{\begin{aligned} \displaystyle-\Delta u&\displaystyle=\lambda u^{q(x)},&% \hskip 10.0ptx&\displaystyle\in\Omega,\\ \displaystyle u&\displaystyle=0,&\hskip 10.0ptx&\displaystyle\in\partial\Omega% ,\end{aligned}\right. where Ω ⊂ ℝ N {\Omega\subset\mathbb{R}^{N}} is a bounded smooth domain, λ > 0 {\lambda>0} is a parameter and the reaction order q ( x ) {q(x)} is a Hölder continuous positive function satisfying q ( x ) > 1 {q(x)>1} for all x ∈ Ω {x\in\Omega} . The relevant feature here is that q is assumed to achieve the value one on ∂ Ω {\partial\Omega} . By assuming that q is subcritical, our main result states the existence of a positive solution for all λ > 0 {\lambda>0} . We also study its asymptotic behavior as λ → 0 {\lambda\to 0} and as λ → ∞ {\lambda\to\infty} . It should be noticed that the fact that q = 1 {q=1} somewhere in ∂ Ω {\partial\Omega} gives rise to serious difficulties when looking for critical points of the functional associated with the problem above. This work is a continuation of [13] where q is assumed to take values both greater and smaller than one in Ω, but is constrained to satisfy q ( x ) > 1 {q(x)>1} on ∂ Ω {\partial\Omega} .