Global regularity results for non-homogeneous growth fractional problems

Abstract

This article concerns with the global Hölder regularity of weak solutions to a class of problems involving the fractional $(p,q)$-Laplacian, denoted by $(-\Delta {)}_p^{s_1}+(-\Delta {)}_q^{s_2}$, for $1<p,q<\infty$ and $s_1,s_2\in (0,1)$. We use a suitable Caccioppoli inequality and local boundedness result in order to prove the weak Harnack type inequality. Consequently, by employing a suitable iteration process, we establish the interior Hölder regularity for local weak solutions, which need not be assumed bounded. The global Hölder regularity result we prove expands and improves the regularity results of Giacomoni, Kumar and Sreenadh (arXiv: 2102.06080) to the subquadratic case (that is, $q<2$) and more general right hand side, which requires a different and new approach. Moreover, we establish a nonlocal Harnack type inequality for weak solutions, which is of independent interest.

References in corpus (10)

• Local behavior of fractional $p$-minimizers
• The second eigenvalue of the fractional $p-$Laplacian
• Nonlocal equations with measure data
• A Hopf's lemma and a strong minimum principle for the fractional $p$-Laplacian
• Regularity results and Harnack inequalities for minimizers and solutions of nonlocal problems: a unified approach via fractional De Giorgi classes
• Regularity estimates for elliptic nonlocal operators
• Multiplicity results for $(p,q)$ fractional elliptic equations involving critical nonlinearities
• Interior and up to the boundary regularity for the fractional $g$-Laplacian: the convex case
• Interior and boundary regularity results for strongly nonhomogeneous $p,q$-fractional problems
• On the logistic equation for the fractional p-Laplacian

Cited by in corpus (5)

• Regularity for nonlocal problems with non-standard growth
• Harnack inequality for nonlocal problems with non-standard growth
• Choquard equation involving mixed local and nonlocal operators
• On generalized eigenvalue problems of fractional $(p,q)$-Laplace operator with two parameters
• Local regularity for nonlocal double phase equations in the Heisenberg group