Harnack inequality for the nonlocal equations with general growth

Abstract

We consider a class of generalized nonlocal p-Laplacian equations. We find some proper structural conditions to establish a version of nonlocal Harnack inequalities of weak solutions to such nonlocal problems by using the expansion of positivity and energy estimates.

Harnack inequality for nonlocal problems with non-standard growth

“…Similar to the approach in this article, they derive local boundedness and a tail estimate as in Theorem 3.1, Lemma 5.1, as well as a weak Harnack inequality. By combining these results, [20] derive an upper estimate for sup u in terms of inf u and a nonlocal tail term. However, for p < q, this result is not optimal due to the appearance of an additional power ι = q/ p in the Harnack inequality.…”

“…Recently, Fang and Zhang have investigated Harnack inequalities for nonlocal operators with general growth [20]. In comparison to our setup, they impose more restrictive structural assumptions on the growth function f .…”

Harnack inequality for nonlocal problems with non-standard growth

“…Similar to the approach in this article, they derive local boundedness and a tail estimate as in Theorem 3.1, Lemma 5.1, as well as a weak Harnack inequality. By combining these results, [FZ22] derive an upper estimate for sup u in terms of inf u and a nonlocal tail term. However, for p < q, this result is not optimal due to the appearance of an additional power ι = q/p in the Harnack inequality.…”

“…Recently, Fang and Zhang have investigated Harnack inequalities for nonlocal operators with general growth [FZ22]. In comparison to our setup, they impose more restrictive structural assumptions on the growth function f .…”