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In this paper, we investigate the existence of solutions for a class of quasilinear elliptic system. By developing the Moser iteration technique, we obtain that system has a nontrivial solution (u λ, v λ) with ‖ (u λ, v λ) ‖ ∞ ≤ 2 for every λ large enough when the nonlinear term F satisfies some growth conditions only in a circle with center 0 and radius 4, and the families of solutions (u λ, v λ) satisfy that ‖ (u λ, v λ) ‖ → 0 as λ → ∞. Moreover, because the interaction of u and v in elliptic system causes that the estimate of ‖ u ‖ ∞ cannot vary with ‖ u ‖, the conclusion for the elliptic system is weaker than the corresponding result for the quasilinear elliptic equation, which is given in the end as a comparison.
Zhang et al. (Wed,) studied this question.