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In this article we show the existence and multiplicity of positive solutions for the nonlocal problem with a sign-changing weight function, $$\displaylines{ -(a-b\int_{\Omega}|\nabla u|^2dx)\Delta u= f_\lambda(x)|u|^{q-2}u, \quad\text{in }\Omega, \cr u=0, \quad\text{on }\partial\Omega, }$$ where $\Omega$ is a smooth bounded domain in $\mathbb{R}^{3}$, $a, b>0,; 1<q<2$. Our technical approach is based on the variational method.