On the higher differentiability of solutions to a class of variational problems of fast growth

We consider the higher differentiability of a solution u to the problem of minimizing $$\begin{aligned} \int _{\Omega }[\Lambda (x ,|\nabla v(x)|) +f(x)v(x)]dx \end{aligned}$$ where \(\Lambda \) is of fast growth in the second variable, i.e., we assume that \(\Lambda (x,t)\) grows in t faster than \(t^N\), where N is the dimension of the space. We do not assume conditions limiting above the size of the second derivative of \(\Lambda \) with respect to t.