Singular solutions for coercive quasilinear elliptic inequalities with nonlocal terms

Abstract

We study the inequality $$ {\rm div}\big(|x|^{-α}|\nabla u|^{m-2}\nabla u\big)\geq (I_β\ast u^p)u^q \quad\mbox{ in } B_1\setminus\{0\}\subset {\mathbb R}^N, $$ where $\alpha >0$, $N\ge 1$, $m>1$, $p,q>m-1$ and $I_{\beta }$ denotes the Riesz potential of order $\beta \in (0,N)$. We obtain sharp conditions in terms of these parameters for which positive singular solutions exist. We further establish the asymptotic profile of singular solutions to the double inequality $$ a(I_β\ast u^p)u^q\geq {\rm div}\big(|x|^{-α}|\nabla u|^{m-2}\nabla u\big)\geq b(I_β\ast u^p)u^q \quad\mbox{ in } B_1\setminus\{0\}\subset {\mathbb R}^N, $$ where $a\ge b>0$ are constants.

26 pages

References in corpus (3)

• A guide to the Choquard equation
• Positive solutions for quasilinear elliptic inequalities and systems with nonlocal terms
• Multiple nonradial solutions for a nonlinear elliptic problem with singular and decaying radial potential