Multiple solutions for a class of quasilinear problems with double criticality

TL;DR

This paper proves the existence of multiple solutions for a class of quasilinear elliptic problems with double critical growth, involving variable operators and nonlinearities that exhibit exponential and polynomial critical behaviors.

Contribution

It introduces new multiplicity results for quasilinear problems with mixed critical nonlinearities and variable operators on complex domains, including nonradial solutions.

Findings

Multiple solutions established for different nonlinearities.

Existence of nonradial, rotationally nonequivalent solutions.

Results applicable to complex domain geometries.

Abstract

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.