Infinitely many solutions to a class of quasilinear Schrodinger system in RN

In this work, we study the existence of multiple solutions to the quasilinear Schrodinger system of k equations -Delta(p)u(j) + a(j)(x)vertical bar u(j)vertical bar(p-2)u(j) = mu(j)vertical bar u(j)vertical bar(q-2)u(j) +1/2 Sigma(i not equal j) beta(ij)vertical bar u(j)vertical bar(m)vertical bar u(j)vertical bar(m-2)u(j), x is an element of R-N, with u(j)(x) -> 0 as vertical bar x vertical bar -> infinity, j = 1,2, ..., k, and N >= 2, 1 < p < N,k >= 2, the potential a(j)(x) is positive and bounded in R-N, mu(j) > 0, beta(ij), = beta(ji) for i not equal 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. (C) 2015 Elsevier Ltd. All rights reserved.