Start ing from a new sum decomposit ion of W1,p(RN ) ∩W1,q(RN ) and using
a variat ional approach, we invest igate the existence of mult ipleweak solut ions of
a (p, q)–Laplacian equat ion on RN , for 1 < q < p < N , with a sign–changing potent
ial and a Carath´eodory react ion term sat isfying the celebrated Ambroset t i–
Rabinowitz condit ion. Our assumpt ions are mild and different from those used
in related papers and moreover our results improve or complement previous ones
for the single p–Laplacian.