January/February 2023 Quasilinear elliptic inequalities with nonlinear convolution terms and potentials of slow decay
Daniel Devine, Marius Ghergu, Paschalis Karageorgis
Differential Integral Equations 36(1/2): 1-20 (January/February 2023). DOI: 10.57262/die036-0102-1

Abstract

We are concerned with the existence and nonexistence of positive weak solutions to $$ -\Delta_m u + \frac{\lambda}{|x|^\gamma}u^{m-1} \geq (K*u^p)u^q \quad\mbox{ in }\mathbb R^N\setminus \overline B_1, $$ where $m > 1$, $N \geq 1$, $\lambda,p > 0$ and $q, \gamma\in\mathbb R$. We assume that $K$ is a positive and continuous function on $\mathbb R^N\setminus \overline B_1$ such that $L(R):=\min_{|x|=R} K(x)$ is decreasing in a neighborhood of infinity. Also, $K\ast u^p$ denotes the standard convolution operation. We obtain conditions on $N,m,\gamma,p$, and $q$ such that the above problem has no solutions. For potentials of slow decay (corresponding to the case $\gamma < m$) and functions $K(x)$ that behave like $|x|^{\alpha-N} (\log |x|)^\beta$ at infinity, these conditions are shown to be essentially optimal.

Citation

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Daniel Devine. Marius Ghergu. Paschalis Karageorgis. "Quasilinear elliptic inequalities with nonlinear convolution terms and potentials of slow decay." Differential Integral Equations 36 (1/2) 1 - 20, January/February 2023. https://doi.org/10.57262/die036-0102-1

Information

Published: January/February 2023
First available in Project Euclid: 12 September 2022

Digital Object Identifier: 10.57262/die036-0102-1

Subjects:
Primary: 35B09 , 35B45 , 35J625

Rights: Copyright © 2023 Khayyam Publishing, Inc.

Vol.36 • No. 1/2 • January/February 2023
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