We consider the mixed local and nonlocal functionals with nonstandard growth \begin{ eqnarray*} u\mapsto\int_{ \Omega}(|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 H\"{ o}lder 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].
Categories: math.AP