We consider a functional of the type $\mathcal{F}(u,\Omega )=\int _{\Omega}F\big(D^{k}u(x)\big)dx$ on the Dirichlet class, where F is a continuous function and Ω is an open bounded set of $\mathbb{R}^{n}$ with a Lipschitz boundary. We prove that coercivity and mean coercivity are equivalent under growth conditions, and further we prove that mean coercivity and quasiconvexity are equivalent. Subsequently, we deduce that $\mathcal{F}(u,\Omega )$ has a minimum under the condition that the integrand F satisfies the growth condition and mean coercivity.
