*Published Paper*

**Inserted:** 1 feb 2002

**Last Updated:** 6 jul 2002

**Journal:** J. Differential Equations

**Volume:** 157

**Number:** 2

**Pages:** 414-438

**Year:** 1999

**Abstract:**

We consider functionals of the type
$$ \int f(Du) +a(x)u$$
defined on vector values functions $u:\Omega \to R^N$,
where $f$ satisfies growth conditions of $(p,q)$ type:
$$ L^{{}-1}

z^{p} \leq f(z) \leq L(1+

z^{q).$$}
Moreover $f$ is convex and satisfies the natural growth and ellipticity assumptions on the matrix
$D^2f$.
We prove local $W^{1,q}$ regularity of minimizers provided
$$ q*p < (n+2)*n$$
where $\Omega \subset R^n$. A higher differentiability result is also given.