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Let $\mathbb{F}_p$ be the field with $p$ elements with $p$ prime,$X_1,\ldots, X_n$ pairwise disjoint subsets of $\mathbb{F}_p$with at least $3$elements such that $\sum_{i=1}^n|X_i|\leq p-5$, and $\mathbb{S}_n$ the set ofpermutations of $\{1,2,\ldots, n\}$. If $a_1,\ldots,a_n\in\mathbb{F}_p^*$ arenot all equal, we characterize the subsets $X_1,\ldots, X_n$ which satisfy\begin{equation*}\Bigg|\bigcup_{\sigma\in\mathbb{S}_n}\sum_{i=1}^na_{\sigma(i)}X_i\Bigg|\leq\sum_{i=1}^n|X_i|. \end{equation*} This result has the following application:For $n\geq 2$, $b\in\mathbb{F}_p$ and $a_1,\ldots, a_n$ as above, wecharacterize the colorings $\bigcup_{i=1}^nC_i=\mathbb{F}_p$ where each colorclass has at least 3 elements such that $\sum_{i=1}^na_ix_i=b$ has not rainbowsolutions.

An inverse theorem in $\mathbb{F}_p$ and rainbow free colorings