### Existence of positive solutions for quasilinear elliptic systems involving the $p$-Laplacian.

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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 $$\left\{\begin{array}{c}{\left(\varphi \left(p\left(t\right){u}^{(n-1)}\right)\left(t\right)\right)}^{\text{'}}+a\left(t\right)f(t,u\left(t\right),{u}^{\text{'}}\left(t\right),...,{u}^{(n-2)}\left(t\right))=0,\phantom{\rule{1.0em}{0ex}}\phantom{\rule{4pt}{0ex}}0<t<1,\hfill \\ {u}^{\left(i\right)}\left(0\right)=0,\phantom{\rule{1.0em}{0ex}}i=0,1,...,n-3,\hfill \\ {u}^{(n-2)}\left(0\right)=\sum _{i=1}^{m-2}{\alpha}_{i}{u}^{(n-2)}\left({\xi}_{i}\right),\phantom{\rule{1.0em}{0ex}}{u}^{(n-1)}\left(1\right)=0,\hfill \end{array}\right.$$ where $\varphi :\mathbb{R}\to \mathbb{R}$ 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.

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