An eigenvalue problem for a nonlocal quasilinear anisotropic equation in fractional Orlicz Sobolev spaces without the $Δ_2$--condition

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 $Δ_2$--condition on the Young functions involved. We show existence of a sequence of eigenpairs $(u_k,λ_k)\to (0,+\infty)$.

Theorems & Definitions (28)

• Lemma 2.1: Gossez, Lemma 1.2
• Lemma 3.2
• proof
• Theorem 3.3
• proof
• Theorem 3.4
• Lemma 3.5
• proof
• Lemma 3.6
• proof