Critical points of non-regular integral functionals

Abstract

We prove the existence of a bounded positive critical point for a class of functionals such as

for $\Omega$ a bounded open set in $\mathbb R^{N}$, $N>2$,$\gamma+2< p < 2N/(N-2)$, $\gamma>0$, $\gamma\neq 1$ and $a(x),\,b(x)$ measurable function satisfying $0<\alpha\leq a(x)\leq \beta$,$0\leq b(x)\leq\beta$ almost everywhere in $\Omega$.