Asymptotically quasiconvex functionals with general growth conditions

We establish local regularity results for minimizers of autonomous vectorial integrals of Calculus of Variations, assuming $\psi$-growth conditions and imposing $\varphi$-quasiconvexity only in an asymptotic sense, both in the sub-quadratic and super-quadratic case. In particular, we obtain $C^{1,\alpha}$ regularity at points $x_0$ where $Du$ is sufficiently large in a neighborhood of $x_0$, as well as Lipschitz regularity on a dense set. \ The results hold for all pairs of Young functions $(\varphi, \psi)$ satisfying the $\Delta_2$ condition.