np complete – NP-completeness of variant SAT: SAT-5Clauses

It is unknown whether the problem is $NP$-complete. In particular, you problem is non-trivial and it is in $P$. Therefore it is $NP$-complete if and only if $P=NP$.

To see that your problem is in $P$ you can consider the following “brute-force” algorithm.
Let $n$ be the number of variables and $n$ be the number of clauses.

If it is possible to simultaneously satisfy $5$ clauses, then there are $5$ literals that are true and belong to different clauses.

You can then explicitly consider all subsets of $min{n, 5}$ variables and, for each subset $S$, try all possible (constantly many) truth assignments to the variables in $S$.

For each of these $O(n^5)$ (partial) truth assignments, check whether at least $5$ clauses are satisfied. This can in constant time after a linear-time preprocessing of the input instance.

The overall time required is then $O(mn + n^5)$ where $O(mn)$ is an upper bound on the size of the instance.