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