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We consider the problem of minimizing multiple integrals of product type, i.e. \[ \min \left\{ \int_\Omega g(u(x))f(\nabla u(x)) dx\colon u\in u_0+W^{1,p}_0 \Omega \right\} \leqno{(\Pcal)} \] where $\Omega$ is a bounded, open set in ${\mathbb R}^N$ , $ f\colon{\mathbb R}^N \to [0, \infty)$ is a possibly nonconvex, lower semicontinuous function with p-growth at infinity for some $1< p < \infty$ and the boundary datum $u_0$ is in $W^{1,p}(\Omega)\cap L^\infty (\Omega)$ (or simply in $W^{1,p}(\Omega)$ if $N< p < \infty$ ). Assuming that the convex envelope $f^{\ast\ast}$ off is affine on each connected component of the set $\{f^{\ast\ast}< f \}$ , we prove attainment for ( ${\cal P$ ) for every continuous, positively bounded below function g such that (i) every point $t\in{\mathbb R}$ is squeezed between two intervals where g is monotone and (ii) g has no strict local minima. This shows in particular that the class of coefficents g that yield existence to ( ${\cal P}$ ) is dense in the space of continuous, positive functions on ${\mathbb R}$ . We present examples which show that these conditions for attainment are essentially sharp. |
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