Summary: We consider a properly immersed submanifold \(M\) in a complete Riemannian manifold \(N\). Assume that the sectional curvature \(K^N\) of \(N\) satisfies \(K^N\geqslant -L(1+\mathrm{dist}_N(\cdot ,q_0)^2)^{\frac{\alpha}{2}}\) for some \(L>0\), \(2>\alpha\geqslant 0\) and \(q_0\in N\). If there exists a positive constant \(k>0\) such that \(\Delta|\mathbf{H}|^2\geqslant k|\mathbf{H}|^4\), then we prove that \(M\) is minimal. We also obtain similar results for totally geodesic submanifolds. Furthermore, we consider a properly immersed submanifold \(M\) in a complete Riemannian manifold \(N\) with \(K^N\geqslant -L(1+\mathrm{dist}_N(\cdot ,q_0)^2)^{\frac{\alpha}{2}}\) for some \(L>0\), \(2>\alpha\geqslant 0\) and \(q_0\in N\). Let \(u\) be a smooth non-negative function on \(M\). If there exists a positive constant \(k>0\) such that \(\Delta u\geqslant ku^2\), and \(|\mathbf{H}|\leqslant C(1+\mathrm{dist}_N(\cdot ,q_0)^2)^{\frac{\beta}{2}}\) for some \(C>0\) and \(1>\beta\geqslant 0\), then we prove that \(u=0\) on \(M\). By using the above result, we show that a non-negative biminimal properly immersed submanifold \(M\) in a complete Riemannian manifold \(N\) with \(0\geqslant K^N\geqslant -L(1+\mathrm{dist}_N(\cdot ,q_0)^2)^{\frac{\alpha}{2}}\) is minimal. These results give affirmative partial answers to the global version of the generalized Chen conjecture for biharmonic submanifolds.