| Summary: | In this paper, we study a nonlinear boundary value system with $p$-Laplacian operator
$$\left\{\begin{array}{lll}
(\phi_{p_1}(u'))'+a_1(t)f(u,v)=0, 0<t<1,\cr
(\phi_{p_2}(v'))'+a_2(t)g(u,v)=0, 0<t<1,\cr
\alpha_1\phi_{p_1}(u(0))-\beta_1\phi_{p_1}(u'(0))=\gamma_1\phi_{p_1}(u(1))+\delta_1\phi_{p_1}(u'(1))=0,
\alpha_2\phi_{p_2}(v(0))-\beta_2\phi_{p_2}(v'(0))=\gamma_2\phi_{p_2}(v(1))+\delta_2\phi_{p_2}(v'(1))=0,
\end{array}\right.$$
where $\phi_{p_i}(s)=|s|^{p_i-2}s, p_i>1, i=1,2$. We obtain some sufficient conditions for the existence of two positive solutions or infinitely many positive solutions by using a fixed-point theorem in cones. Especially, the nonlinear terms $f,g $ are allowed to change sign. The conclusions essentially extend and improve the known results.
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