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In this paper, we study the following modified quasilinear fourth-order elliptic systems  42u−4u+ (λα(x) + 1)u− 1 2 4(u2)u = p p+ q |u|p−2|v|qu, in R, 42v−4v+ (λβ(x) + 1)v− 1 2 4(v2)v = q p+ q |u|p|v|q−2v, in R, where 42 = 4(4) is the biharmonic operator, λ > 0, and 2 < p, 2 < q, 4 < p+ q < 22∗∗, 2∗∗ = 2N N−4 (N 6 5) (if N 6 4, 2 ∗∗ =∞) is the critical Sobolev exponent for the embedding W2,2(RN) ↪→ L2(RN). Under some appropriate assumptions on α(x) and β(x), we obtain that the above problem has nontrivial ground state solutions via the variational methods. We also explore the phenomenon of concentration of solutions.

Ground states solutions for modified fourth-order elliptic systems with steep well potential