On certain diophantine equations of diagonal type

In this note we consider diophantine equations of the forma(xp−yq)=b(zr−ws),where 1p+1q+1r+1s=1, with even positive integers p, q, r, s  . We show that in each case the set of rational points on the underlying surface is dense in the Zariski topology. For the surface with (p,q,r,s)=(2,6,6,6)(p,q,r,s)=(2,6,6,6) we prove density of rational points in the Euclidean topology. Moreover, in this case we construct infinitely many parametric solutions in coprime polynomials. The same result is true for (p,q,r,s)∈{(2,4,8,8),(2,8,4,8)}(p,q,r,s)∈{(2,4,8,8),(2,8,4,8)}. In the case (p,q,r,s)=(4,4,4,4)(p,q,r,s)=(4,4,4,4), we present some new parametric solutions of the equation x4−y4=4(z4−w4)x4−y4=4(z4−w4).