In this work, we study the existence of multiple solutions to the quasilinear Schrödinger system of
k equations
with
uj(x)→0 as
|x|→∞,j=1,2,…,k, and
N≥2,1<p<N,k≥2, the potential
aj(x) is positive and bounded in
RN,
渭j>0,尾ij=尾ji for
i≠j,j=1,…,k. We develop a new technique to verify the
(PS) condition and then apply a version of mountain pass lemma to prove the existence of infinitely many nonnegative solutions to the above system.