  We establish multiplicity results for the following class of quasilinear
problems $$ \left\{ \begin{array}{l} -\Delta_{\Phi}u=f(x,u) \quad \mbox{in}
\quad \Omega, \\ u=0 \quad \mbox{on} \quad \partial \Omega, \end{array} \right.
\leqno{(P)} $$ where $\Delta_{\Phi}u=\text{div}(\varphi(x,|\nabla u|)\nabla u)$
for a generalized N-function $\Phi(x,t)=\int_{0}^{|t|}\varphi(x,s)s\,ds$. We
consider $\Omega\subset\mathbb{R}^N$ to be a smooth bounded domain that
contains two disjoint open regions $\Omega_N$ and $\Omega_p$ such that
$\overline{\Omega_N}\cap\overline{\Omega_p}=\emptyset$. The main feature of the
problem $(P)$ is that the operator $-\Delta_{\Phi}$ behaves like $-\Delta_N$ on
$\Omega_N$ and $-\Delta_p$ on $\Omega_p$. We assume the nonlinearity
$f:\Omega\times\mathbb{R}\to\mathbb{R}$ of two different types, but both
behaves like $e^{\alpha|t|^\frac{N}{N-1}}$ on $\Omega_N$ and $|t|^{p^*-2}t$ on
$\Omega_p$ as $|t|$ is large enough, for some $\alpha>0$ and
$p^*=\frac{Np}{N-p}$ being the critical Sobolev exponent for $1<p<N$. In this
context, for one type of nonlinearity $f$, we provide multiplicity of solutions
in a general smooth bounded domain and for another type of nonlinearity $f$, in
an annular domain $\Omega$, we establish existence of multiple solutions for
the problem $(P)$ that are nonradial and rotationally nonequivalent.
