Consider the following 3-SAT variant defined over the variables $x_1,\ldots,x_n$. In the $k$P$k$N-3SAT problem each variable $x_j$, $j \in [n]$, occurs exactly $k$ times as a positive literal in $\phi$, and exactly $k$ times as a negative literal in $\phi$, where $\phi$ is a 3-CNF formula. The problem is then to decide if such a formula has a satisfying assignment.
Is the $k$P$k$N-3SAT problem NP-complete?
In the $m$P$n$N-SAT problem each positive literal occurs exactly $m$ times in $\phi$, and each negative literal occurs exactly $m$ times in $\phi$, where $\phi$ is a CNF formula. It was shown in  that $2$P$1$N-SAT is NP-complete. This hints that the $k$P$k$N-3SAT problem is hard as well.
The $1$P$1$N-SAT is apparently easy, see a related question and answer here. Is $k$P$k$N-3SAT perhaps hard already for $k \geq 2$?
 Yoshinaka, Ryo. "Higher-order matching in the linear lambda calculus in the absence of constants is NP-complete." Term Rewriting and Applications. Springer Berlin Heidelberg, 2005. 235-249.