Local behaviour of the mixed local and nonlocal problems with nonstandard growth

Abstract

We consider the mixed local and nonlocal functionals with nonstandard growth \begin{eqnarray*} u\mapsto\int_Ω(|Du|^p-f(x)u)\,dx+\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{|u(x)-u(y)|^q}{|x-y|^{N+sq}}\,dxdy \end{eqnarray*} with $1<p\le sq$, $0<s<1$ and $\Omega \subset {\mathbb{R}}^N$ being a bounded domain. We study, by means of expansion of positivity, local behaviour of the minimizers of such problems, involving local boundedness, local Hölder continuity and Harnack inequality. The results above can be seen as a natural extension of the results under the center condition that $p>sq$ in [De Filippis-Mingione, Math. Ann., https://doi.org/10.1007/s00208-022-02512-7].