We show that for each $\lambda > 0$, the problem
$-\Delta_p u = \lambda f(u)$ in $Omega$,
$u = 0$ on $\partial \Omega$
has a sequence of positive solutions $(u_n)_n$
with $\max_{\bar\Omega} u_n$ decreasing to zero.
We assume that $\displaystyle{\liminf_{s\to0^+}\frac{F(s)}{s^p} = 0}$
and that
$\displaystyle{\limsup_{s\to 0^+}\frac{F(s)}{s^p} = +\infty}$,
where $F'=f$. We stress that no condition on the sign of $f$ is imposed.
