Supersolutions to nonautonomous Choquard equations in general domains

Abstract

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.