Existence and boundedness of minimizers of a class of integral functionals

Mercaldo, A.

Bollettino della Unione matematica italiana, Série 8, 6B (2003) no. 1, pp. 125-139

Résumé

In this paper we consider a class of integral functionals whose integrand satisfies growth conditions of the type \begin{gather*} f(x, \eta, \xi) \geq a(x) \frac{|\xi|^{p}}{(1 + |\eta|)^{\alpha}} - b_{1}(x)|\eta|^{\beta_{1}}-g_{1}(x),\\ f(x, \eta, 0)\leq b_{2}(x)|\eta|^{\beta_{2}}+ g_{2}(x), \end{gather*} where $0\leq \alpha $, $1\leq \beta_{1} p$, $0\leq \beta_{2} p$, $\alpha+\beta_{i}\leq p$, $a(x)$, $b_{i}(x)$, $g_{i}(x)$ ($i= 1$, $2$) are nonnegative functions satisfying suitable summability assumptions. We prove the existence and boundedness of minimizers of such a functional in the class of functions belonging to the weighted Sobolev space $W^{1,p}(a)$ , which assume a boundary datum $u_{0}\in W^{1,p}(a)\cap L^{\infty}(\Omega)$.