Abstract
Let \Omega be a \mathcal{C}^{2} bounded domain of {{\mathbb R}}^N, N\geq
2. We consider the following quasilinear elliptic problem: ({
P}_{\lambda})\left\lbrace \begin{array}{l} -\Delta_p u = K(x)(\lambda
u^q-u^r),\quad \ \ \mbox{ in }\Omega, \\ \quad \;\;\;u= 0 \quad\mbox{ on
}\partial\Omega, \quad u\geq 0\quad\mbox{ in }\Omega, \end{array}\right.
where p>1 and \Delta_p u{\stackrel{{\rm {def}}}{=}} \mathrm{div}
\left(\vert \nabla u\vert ^{p-2}\nabla u\right) denotes the p-Laplacian
operator. In this paper, \lambda>0 is a real parameter, the exponents q
and r satisfy -1<r<q<p-1, and K:\Omega\longrightarrow {{\mathbb
R}} is a positive function having a singular behaviour near the boundary
\partial \Omega. Precisely, K(x)=d(x)^{-k}L(d(x)) in \Omega, with
0<k<p, L a positive perturbation function, and d(x) the distance of
x\in\Omega to \partial\Omega. By using a sub- and supersolution technique,
we discuss the existence of positive solutions or compact support solutions of
({ P}_{\lambda}) in respect to the blow-up rate k. Precisely, we prove that
if k<1+r, ({ P}_{\lambda}) has at least one positive solution for
\lambda>0 large enough, whereas it has only compact support solutions if
k\geq 1+r.