On singular problems in nonreflexive fractional Orlicz-Sobolev spaces

TL;DR

This paper establishes existence, uniqueness, and convergence of positive solutions for singular fractional quasilinear problems in nonreflexive Orlicz-Sobolev spaces, overcoming challenges posed by nonreflexivity and singularity.

Contribution

It introduces a new approach for constructing test functions and proves the existence of positive solutions in nonreflexive fractional Orlicz-Sobolev spaces.

Findings

Existence and uniqueness of positive solutions for the singular problem.

Convergence of solutions as the fractional parameter approaches 1.

A novel method for constructing test functions in nonreflexive spaces.

Abstract

In this work, we deal with existence and uniqueness of positive solution $u_s$ for the singular quasilinear problem $(-\Delta_{\Phi})^su=u^{-\gamma}$ in the nonreflexive fractional Orlicz-Sobolev $ W^{s}_0L^{\Phi}(\Omega)$ for $0<s<1$. Furthermore, we show that $u_s$ converges in $L^{\Phi}(\Omega)$ to the unique positive solution $u\in W^{1}_0L^{\Phi}(\Omega)$ of the problem $-\Delta_{\Psi}u=u^{-\gamma}$ as $s \uparrow 1$, where $\Psi$ is an appropriate $N$-function equivalent to the $N$-function $\Phi$. The main difficulties to obtain existence of weak solutions for both singular quasilinear problems are that their associate energy functionals may not be well-defined on their whole natural workspaces due to the lack of the reflexivity and the presence of the singular term. To overcome these difficulties, we will use the minimization method and present a new approach to building appropriate test functions to prove that the problems have positive minimizers that we showed to be weak solutions of them, respectively.