Multiple solutions for a class of quasilinear problems with double criticality

Abstract

AbstractWe 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.