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In this paper, we study the multiplicity of solutions for a class of noncooperative -Laplacian operator elliptic system. Under suitable assumptions, we obtain a sequence of solutions by using the limit index theory.

The paper deals with the existence of multiple positive solutions for the boundary value problem $$\left\{\begin{array}{c}{\left(\varphi \left(p\left(t\right)u^{(n-1)}\right)\left(t\right)\right)}^{\text{'}}+a\left(t\right)f(t,u\left(t\right),u^{\text{'}}\left(t\right),...,u^{(n-2)}\left(t\right))=0,0<t<1,\hfill \\ u^{\left(i\right)}\left(0\right)=0,i=0,1,...,n-3,\hfill \\ u^{(n-2)}\left(0\right)=\sum _{i=1}^{m-2}{\alpha }_i{u}^{(n-2)}\left({\xi }_i\right),u^{(n-1)}\left(1\right)=0,\hfill \end{array}\right.$$ where $\varphi :ℝ\to ℝ$ is an increasing homeomorphism and a positive homomorphism with $\varphi \left(0\right)=0$. Using a fixed-point theorem for operators on a cone, we provide sufficient conditions for the existence of multiple positive solutions to the above boundary value problem.