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(1u)}^{p}, p > 1 p\gt 1. We also discuss the Liouvilletype results in unbounded domains.
Original language  English 

Article number  20230107 
Journal  Advances in Nonlinear Analysis 
Volume  12 
Issue number  1 
DOIs  
Publication status  Published  2023 
MoE publication type  A1 Journal articlerefereed 
Keywords
 eigenvalue problems
 Liouvilletype theorems
 mLaplace operator
 quasilinear elliptic equations
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New Nonlinear Analysis Study Results from Aalto University Described (Supersolutions To Nonautonomous Choquard Equations In General Domains)
22/12/2023
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