On the basis of some new Liouville theorems, under suitable conditions, a priori estimates are obtained of positive solutions of the problem \[-\Delta _pu=\lambda u^{\alpha }-a(x)u^q\quad \mbox{in}\;\Omega,\qquad u|_{\partial \Omega }=0,\] where $\Omega \subset {\mathbb{R}}^N$ ($N\geq 2$) is a bounded smooth domain, $p>1$ and λ is a parameter, α, q are given constants such that $p-1<\alpha <p^*-1$, $\alpha <q$, $p^*=Np/(N-p)$ if $N > p$ and $p^*=\infty $ when $N\leq p$, and $a(x)$ is a continuous nonnegative function. Making use of the Leray–Schauder degree of a compact mapping and a priori estimates, the paper finds that the problem above possesses at least one positive solution. It also discusses the corresponding perturbed problem, where $a(x)$ is replaced by $a(x)+\epsilon$, $\epsilon>0$. The results are strikingly different from those obtained for the case $\alpha=p-1$.