Critical points of non-regular integral functionals

We prove the existence of a bounded positive critical point for a class of functionals such as $$ J(v)=\frac12\io [a(x)+b(x)|v|^{\gamma}]|\nabla v|^{2}-\io |v|^{p} $$ for $\Omega$ a bounded open set in $\R^{N}$, $N>2$, $\gamma+2<p<2N/(N-2)$, $\gamma>0$, $\gamma\neq 1$ and $a(x),\,b(x)$ measurable function satisfying $0<\alpha\leq a(x)\leq \beta$, $0\leq b(x)\leq\beta$ almost everywhere in $\Omega$.