Summary: Let \(\Omega\) be a bounded domain in \(\mathbb{R}^N\) \((N>2)\). We are concerned with the existence and nonexistence of solutions for the following nonlocal problem \[ -M\Biggl(\int_\Omega |\nabla u(x)|^2dx\Biggr)\,\Delta u= |u|^{p-1} u+\lambda f(x)\quad\text{in }\Omega,\;u|_{\partial\Omega}= 0, \] where \(M\) is a continuous function on \(\mathbb{R}^+\) and \(f\in C^1(\overline\Omega)\) changes sign. \(\lambda\) and \(p\) are positive parameters. By direct variational methods, the Galerkin approach and sub- and super-solutions methods, some results are established.