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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 $\Omega\subset\mathbb{R}^N$ beinga bounded domain. We study, by means of expansion of positivity, localbehaviour of the minimizers of such problems, involving local boundedness,local H\"{o}lder continuity and Harnack inequality. The results above can beseen 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].

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