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We establish multiplicity results for the following class of quasilinear problems
P\begin{equation*} \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. \end{equation*}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 behave 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 a 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 non-radial and rotationally non-equivalent.
