Quasilinear elliptic inequalities with nonlinear convolution terms and potentials of slow decay

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.