Supersolutions to nonautonomous Choquard equations in general domains

Abstract

We consider the nonlocal quasilinear elliptic problem: −Δmu(x)=H(x)((Iα*(Qf(u)))(x))βg(u(x))inΩ,-{\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 RN{{\mathbb{R}}}^{N}, β≥0\beta \ge 0, Iα{I}_{\alpha }, 00f\left(s),g\left(s)\gt 0 for s>0s\gt 0, and H,Q:Ω→RH,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 ff and gg such as eu,(1+u)p{e}^{u},{\left(1+u)}^{p}, and (1−u)−p{\left(1-u)}^{-p}, p>1p\gt 1. We also discuss the Liouville-type results in unbounded domains.

Keywords

• quasilinear elliptic equations
• m-laplace operator
• liouville-type theorems
• eigenvalue problems
• 35j92
• 35a23
• 35b09
• 35b53
• 47j10