Multiple solutions for a class of quasilinear problems with double criticality
We establish multiplicity results for the following class of quasilinear problems $$ \left\{ \begin{array}{l} -Δ_Φu=f(x,u) \quad \mbox{in} \quad Ω, \\ u=0 \quad \mbox{on} \quad \partial Ω, \end{array} \right. \leqno{(P)} $$ where $Δ_Φu=\text{div}(φ(x,|\nabla u|)\nabla u)$ for a generalized N-function $Φ(x,t)=\int_{0}^{|t|}φ(x,s)s\,ds$. We consider $Ω\subset\mathbb{R}^N$ to be a smooth bounded domain that contains two disjoint open regions $Ω_N$ and $Ω_p$ such that $\overline{Ω_N}\cap\overline{Ω_p}=\emptyset$. The main feature of the problem $(P)$ is that the operator $-Δ_Φ$ behaves like $-Δ_N$ on $Ω_N$ and $-Δ_p$ on $Ω_p$. We assume the nonlinearity $f:Ω\times\mathbb{R}\to\mathbb{R}$ of two different types, but both behaves like $e^{α|t|^\frac{N}{N-1}}$ on $Ω_N$ and $|t|^{p^*-2}t$ on $Ω_p$ as $|t|$ is large enough, for some $α>0$ 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 $Ω$, we establish existence of multiple solutions for the problem $(P)$ that are nonradial and rotationally nonequivalent.