Boundedness of minimizers of degenerate functionals (Q1909697)

The author considers the functional \(F(u)=\int_\Omega f(x,u,\nabla u)dx\), where \(\Omega\subset\mathbb{R}^n\) \((n\geq2)\) is an open set with finite measure, \(u:\Omega\to\mathbb{R}\), while \(f(x,u,\xi)\) is a real-valued function satisfying the following growth conditions: \(f(x,u,\xi)\geq a(x)H(|u|)|\xi|^p\) and \(f(x,u,0)\leq b(x)B(|u|)\) for a.e. \(x\in\Omega\), \(\forall u\in \mathbb{R}\), \(\forall\xi\in\mathbb{R}^n\), with \(p\geq 1\), \(a\), \(b\) nonnegative measurable functions, \(H\), \(B\) increasing functions in \([0,\infty)\) and \(H(0)>0\). Several conditions on \(a\), \(b\), \(H\), \(B\) are given which assure that a global minimizer of \(F\) on \(W^{1,p}_0(a)\) is bounded.