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

Abstract

We study positive weak super-solutions of the Lane-Emden inequality $\mathcal{L}u := \sum_{i=1}^{n}(-\Delta_{p_i})^{s_i}u \ge u^q$ in $\mathbb{R}^N$, where each $(-\Delta_{p_i})^{s_i}$ is a fractional $p_i$-Laplacian of order $s_i\in(0,1]$ and exponent $p_i\in(1,\infty)$, and $\max_i s_ip_i<N$. Let $\kappa_i := (N-s_ip_i)/(p_i-1)$ be the decay exponent of the fundamental solution of the $i$-th operator, and let $\overline{\kappa} := \max_i\kappa_i$. We show that the Serrin exponent of $\mathcal{L}$ is $\mathfrak{q}_{\mathcal{L}} = N/\overline{\kappa}$. 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 $A(\lambda+|x|)^{-\kappa}$. In other words, the Serrin exponent of the sum is the smallest of the Serrin exponents $N/\kappa_i$ 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.