*Accepted Paper*

**Inserted:** 22 dec 2014

**Last Updated:** 14 oct 2016

**Journal:** J. Optim. Theory Appl.

**Year:** 2015

**Abstract:**

The energy-integral of the calculus of variations \[\mathcal{F}(u;\Omega):=\int_{\Omega }f(x,u,Du(x))\,dx,
\]
where $\Omega$ is an open bounded subset of $\mathbb{R}^n$, $n\ge 2$, and $f: \Omega\times \mathbb{R}\times \mathbb{R}^{n}\rightarrow
\mathbb{R}$ is a Carath\'{e}odory function, $f(x,s,\xi)$ convex in $(s,\xi)$ and satisfying the growth condition
\[
c_1\sum_{i=1}^n

\xi_i

^{p_i} \le {f}(x,s,\xi)\le c_2\left\{1+

s

^q+

\xi

^{q}\right\},
\]
has a
limit behavior when $q=n\overline{p}/(n-\overline{p})$, where $\overline{p}$ is the harmonic average
of the exponents $p_i$, $i=1,\ldots,n$. In fact, if $q$ is larger than in the stated equality, counterexamples to
the local boundedness and regularity of minimizers
are known. In this paper we prove the local boundedness of minimizers (and also of quasi-minimizers)
under this stated limit condition. Some other general and limit growth conditions are also considered.

**Keywords:**
local boundedness, quasi-minimizer, anisotropic growth condition