The Serrin exponent for Lane-Emden inequalities driven by sums of fractional p-Laplacians

We study positive weak super-solutions of the Lane-Emden inequality $\mathcal{L}u:={\sum }_{i=1}^n(-{\mathrm{\Delta }}_{p_i}{)}^{s_i}u\ge u^q$ in ${\mathbb{R}}^N$, where each $(-{\mathrm{\Delta }}_{p_i}{)}^{s_i}$ is a fractional $p_i$-Laplacian of order $s_i\in (0,1]$ and exponent $p_i\in (1,\mathrm{\infty })$, and ${\max }_i{s}_i{p}_i<N$. Let ${\kappa }_i:=(N-s_i{p}_i)\mathrm{/}(p_i-1)$ be the decay exponent of the fundamental solution of the $i$-th operator, and let $\bar{\kappa }:={\max }_i{\kappa }_i$. We show that the Serrin exponent of $\mathcal{L}$ is ${\mathfrak{q}}_{\mathcal{L}}=N\mathrm{/}\bar{\kappa }$. More precisely, for $0<q<{\mathfrak{q}}_{\mathcal{L}}$ the only non-negative continuous weak super-solution is $u\equiv 0$, while for every $q>{\mathfrak{q}}_{\mathcal{L}}$ we construct explicit positive weak super-solutions of the form $A(\lambda +\mathrm{\mid }x\mathrm{\mid }{)}^{-\kappa }$. In other words, the Serrin exponent of the sum is the smallest of the Serrin exponents $N\mathrm{/}{\kappa }_i$ of the single operators, that is, the one of the operator whose fundamental solution decays fastest. No structural assumption relating the different operators is needed. The endpoint $q={\mathfrak{q}}_{\mathcal{L}}$ is left open. The local order $s_i=1$ is treated together with the nonlocal orders, and all the arguments are carried out in the weak setting.