Multiple solutions for a class of quasilinear problems with double criticality

Abstract

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 $\hat{I}{”}_{\hat{I}}¦u=\text{div}(\ddot{I}†(x,|\nabla u|)\nabla u)$ for a generalized N-function $\hat{I}¦(x,t)={\int }_0^{|t|}\ddot{I}†(x,s)sds$. We consider $\hat{I}\text{c◯}\subset {\mathbb{R}}^N$ to be a smooth bounded domain that contains two disjoint open regions $\hat{I}{\text{c◯}}_N$ and $\hat{I}{\text{c◯}}_p$ such that $\overline{\hat{I}{\text{c◯}}_N}\cap \overline{\hat{I}{\text{c◯}}_p}=\varnothing$. The main feature of the problem $(P)$ is that the operator $-\hat{I}{”}_{\hat{I}}¦$ behaves like $-\hat{I}{”}_N$ on $\hat{I}{\text{c◯}}_N$ and $-\hat{I}{”}_p$ on $\hat{I}{\text{c◯}}_p$. We assume the nonlinearity $f:\hat{I}\text{c◯}\times \mathbb{R}\to \mathbb{R}$ of two different types, but both behaves like $e^{\hat{I}\pm |t{|}^{\frac{N}{N-1}}}$ on $\hat{I}{\text{c◯}}_N$ and $|t{|}^{p^{\ast }-2}t$ on $\hat{I}{\text{c◯}}_p$ as $|t|$ is large enough, for some $\hat{I}\pm >0$ and $p^{\ast }=\frac{Np}{N-p}$ 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 $\hat{I}\text{c◯}$, we establish existence of multiple solutions for the problem $(P)$ that are nonradial and rotationally nonequivalent.

30 pages, comments are welcome