Nature-inspired computation is receiving increasing attention. Various Ising machine implementations have recently been proven to be effective in solving numerous combinatorial optimization problems including maximum cut, low density parity check (LDPC) decoding, and Boolean satisfiability (SAT) problems. In this paper, a novel method is presented to solve SAT or MAX-SAT problems with a CMOS circuit implementation. The technique solves a SAT problem by mapping the SAT variables onto quantized capacitor voltages generated by an array of nodes that interact through a network of coupling units. The nodal interaction is achieved through coupling currents produced by the coupling units, which charge or discharge capacitor voltages, implementing a gradient descent along the SAT problem's cost function to minimize the number of unsatisfied clauses. The system also incorporates a unique low-complexity perturbation scheme to avoid settling in local minima, greatly enhancing the performance of the system. The simulation results demonstrate that the proposed SKI-SAT is a high-performance and low-energy alternative that surpasses existing solvers by significant margins, achieving more than 10 times faster solution and 300 times less power.