Infinitely many solutions to a class of quasilinear Schrödinger system in RNRN

In this work, we study the existence of multiple solutions to the quasilinear Schrödinger system of kk equations −Δpuj+aj(x)|uj|p−2uj=μj|uj|q−2uj+12∑i≠jβij|ui|m|uj|m−2uj,x∈RN, with uj(x)→0uj(x)→0 as |x|→∞,j=1,2,…,k|x|→∞,j=1,2,…,k, and N≥2,10,βij=βjiμj>0,βij=βji for i≠j,j=1,…,ki≠j,j=1,…,k. We develop a new technique to verify the (PS)(PS) condition and then apply a version of mountain pass lemma to prove the existence of infinitely many nonnegative solutions to the above system.