  We discuss the existence of multiple positive solutions leading to the
occurrence of an S-shaped bifurcation curve to the equations of the form $$
-\Delta_p u= f(\mu,\lambda, u)~ \mbox{in} ~\Omega \subset \mathbb{R}^N $$ where
$\Delta_p$ is a $p$-Laplacian, $p>1$, $N\geq 1$, $\mu, \lambda \in \mathbb{R}$.
We deal with relatively unexplored cases when $f(\mu,\lambda, u)$ is
non-Lipschitz at $u=0$, $f(\mu,\lambda, 0) = 0 $ and $ f(\mu,\lambda, u) <0$,
$u \in (0,r)$, for some $r<+\infty$. We develop the nonlinear generalized
Rayleigh quotients method to find a range of parameters where the equation may
have distinct branches of positive solutions. As a consequence, applying the
Nehari manifold method and the mountain pass theorem, we prove that the
equation for some range of values $\mu, \lambda$, has at least three positive
solutions with two linearly unstable solutions and one linearly stable.
  The results evidence that the bifurcation curve is S-shaped and exhibits the
so-called dual cusp catastrophe which is characterized by the fact that the
corresponding dynamic equation has stable states only within the cusp-shaped
region in the control plane of parameters. Our results are new even in the
one-dimensional case and $p=2$.
