An eigenvalue problem for a nonlocal quasilinear anisotropic equation in fractional Orlicz Sobolev spaces without the $\Delta_2$--condition

TL;DR

This paper investigates a nonlocal eigenvalue problem involving fractional operators in fractional Orlicz Sobolev spaces without the $ ext{ extDelta}_2$--condition, establishing the existence of an unbounded sequence of eigenpairs.

Contribution

It introduces a novel analysis of a nonlocal eigenvalue problem in fractional Orlicz Sobolev spaces without the $ ext{ extDelta}_2$--condition, extending previous frameworks.

Findings

Existence of a sequence of eigenpairs $(u_k, \lambda_k)$ diverging to infinity.

Generalization of eigenvalue problems to nonlocal, nonstandard growth diffusion models.

Analysis conducted without assuming the $ ext{ extDelta}_2$--condition on Young functions.

Abstract

In this paper we analyze an eigenvalue problem associated to fractional operators of the form \[ L_a^s u(x)=2 \text{p.v.}\int_{\mathbb{R}^n}a(x,y,D^su(x,y))\,\frac{dy}{|x-y|^{n+s}},\] which represents a generalization model for nonlocal, nonstandard growth diffusion problems. We study this problem in the context of the fractional Orlicz Sobolev spaces without assuming the so-called $\Delta_2$--condition on the Young functions involved. We show existence of a sequence of eigenpairs $(u_k,\lambda_k)\to (0,+\infty)$.