In this paper, we deal with the existence of solution for a class of quasilinear Schrödinger equations with a nonlocal term ( ( ) ) ( ) ( ) ( ) ( ( )) ( ) 2 2 3 div , , g u u g u g u u V x u x KF u Kf u x −μ − ∇ + ′ ∇ + = ∗ ∈ where μ ∈(0,3) , the function K,V ∈C(3 ,+ ) and V (x) may be vanish at infinity, g is a C1 even function with g′(t ) ≤ 0 for all t ≥ 0 , g (0) = 1, lim ( ) t g t a →+∞ = , 0 < a < 1 , and F is the primitive function of f which is superlinear but subcritical at infinity in the sense of Hardy-littlewood- Sobolev inequality. By the mountain pass theorem, we prove that the above equation has a nontrivial solution.
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