  In studying rational points on elliptic K3 surfaces of the form
$f(t)y^2=g(x)$, where $f,g$ are cubic or quartic polynomials (without repeated
roots), we introduce a condition on the quadratic twists of two elliptic curves
having simultaneously positive Mordell-Weil rank. We prove a necessary and
sufficient condition for the Zariski density of rational points by using this
condition, and we relate it to the Hilbert property. Applying to surfaces of
Cassels-Schinzel type, we prove unconditionally that rational points are dense
both in Zariski topology and in real topology.
