Double phase inequalities with convolution nonlinearity in exterior domains

Abstract

We discuss the existence of $C^1$-solutions for two related double phase inequalities: \begin{equation*} {\mathcal L}_g u\pm Δ_s u\geq (|x|^{-α}*u^p)u^q \quad\mbox{ in }\mathbb R^N\setminus \overline B_1, N\geq 1,\tag{$P^{\pm }$} \end{equation*} in which $\hat{I}{”}_{s}u:=\mathrm{div}(|\nabla u{|}^{s-2}\nabla u)$ is the $s$-Laplace operator, $s>1$, and $${\mathcal{L}}_{g}u:=-\mathrm{div}(|\nabla u{|}^{m-2}g(|\nabla u|)\nabla u),m>s>1,$$ where $g:[0,\infty )\to (0,\infty )$ is a $C^1(0,\infty )\cap C[0,\infty )$ non-increasing function with some specific behaviour near the origin. In the above context, the general form of ${\mathcal{L}}_{g}u$ includes the case of $m$-Laplace and $m$-mean curvature operator. Our study reveals a sharp distinction between $(P^{+})$ and $(P^{-})$. Precisely, we show that the inequality $(P^{+})$ has solutions for all $m>s>1$ and $q>s-1$. In contrast, $(P^{-})$ has solutions if and only if $p$ and $q$ are sufficiently large. We also link the solvability of $(P^{-})$ with that of the corresponding equation ${\mathcal{L}}_{g}u-\hat{I}{”}_{s}u=(|x{|}^{-\hat{I}\pm }\ast u^p)u^q$ in ${\mathbb{R}}^N\setminus {\overline{B}}_1$, for which we derive optimal conditions in terms of $p,q,\hat{I}\pm ,s$ and $N$. The approach combines integral estimates with a new sub and supersolution method that accounts for the presence of the convolution term.