On passage to the limit in nonlinear variational problems

A study is made of variational problems with convex Lagrangians $f(x,\xi)$ subordinate to a nonstandard estimate
\begin{gather*} -c_0+c_1|\xi|^{\alpha_1}\leqslant f(x,\xi)\leqslant c_0+c_2|\xi|^{\alpha_2}, \\ c_0\geqslant 0, c_1>0, \quad c_2>0, \quad 1<\alpha_1\leqslant\alpha_2. \end{gather*}
The concepts of $\Gamma_1$-convergence and $\Gamma_2$-convergence are introduced for Lagrangians corresponding to boundary value problems of the first and second types. It is proved that the indicated class of Lagrangians is compact with respect to $\Gamma_1$-convergence and with respect to $\Gamma_2$-convergence. Applications to compactness theorems and to various concrete averaging problems are given.