Let \(f(x,\xi)\) be a convex Lagrangian subjected to the estimate \(-c_ 0+c_ 1|\xi|^{\alpha_ 1} \leq f(x,\xi)\leq c_ 0+c_ 2|\xi |^{\alpha_ 2}\), \(c_ 0\geq 0\), \(c_ 1>0\), \(c_ 2>0\), \(1<\alpha_ 1<\alpha_ 2\). The author introduces the notions of \(\Gamma_ 1\)- and \(\Gamma_ 2\)-convergence, which correspond to two types of boundary value problems. For these convergences the compactness theorems are proved. Some applications to homogenization problems are given.