Supersolutions to nonautonomous Choquard equations in general domains

Asadollah Aghajani, Juha Kinnunen*

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review

1 Citation (Scopus)
3 Downloads (Pure)

Abstract

We consider the nonlocal quasilinear elliptic problem: - Δ m u (x) = H (x) ((I α ∗ (Q f (u))) (x)) β g (u (x)) in ω, -{\Delta }_{m}u\left(x)=H\left(x){(\left({I}_{\alpha }∗ \left(Qf\left(u)))\left(x))}^{\beta }g\left(u\left(x))\hspace{1.0em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega, where ω \Omega is a smooth domain in R N {{\mathbb{R}}}^{N}, β ≥ 0 \beta \ge 0, I α {I}_{\alpha }, 0 < α < N 0\lt \alpha \lt N, stands for the Riesz potential, f, g: [ 0, a) → [ 0, ∞) f,g:\left[0,a)\to \left[0,\infty), 0 < a ≤ ∞ 0\lt a\le \infty, are monotone nondecreasing functions with f (s), g (s) > 0 f\left(s),g\left(s)\gt 0 for s > 0 s\gt 0, and H, Q: ω → R H,Q:\Omega \to {\mathbb{R}} are nonnegative measurable functions. We provide explicit quantitative pointwise estimates on positive weak supersolutions. As an application, we obtain bounds on extremal parameters of the related nonlinear eigenvalue problems in bounded domains for various nonlinearities f f and g g such as e u, (1 + u) p {e}^{u},{\left(1+u)}^{p}, and (1 - u) - p {\left(1-u)}^{-p}, p > 1 p\gt 1. We also discuss the Liouville-type results in unbounded domains.

Original languageEnglish
Article number20230107
JournalAdvances in Nonlinear Analysis
Volume12
Issue number1
DOIs
Publication statusPublished - 2023
MoE publication typeA1 Journal article-refereed

Keywords

  • eigenvalue problems
  • Liouville-type theorems
  • m-Laplace operator
  • quasilinear elliptic equations

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