Two Generalizations of Stampacchia Lemma and Applications

We present two generalizations of the classical Stampacchia Lemma which contain a non-decreasing non-negative function $g$, and give applications. As a first application, we deal with variational integrals of the form $$\mathcal{J}(u;\mathrm{\Omega })={\int }_{\mathrm{\Omega }}f(x,Du(x))dx\mathrm{.}$$ We consider a minimizer $u:\mathrm{\Omega }\subset {\mathbb{R}}^n\to \mathbb{R}$ among all functions with a fixed boundary value $u_{\ast }$ on $\mathrm{\partial }\mathrm{\Omega }$. Under some nonstandard growth conditions of the integrand $f(x,\xi )$ we derive some regularity results; as a second application, we consider elliptic equations of the form $$\{\begin{array}{lll}-\text{mbox}div(a(x,u(x))Du(x))=f(x),& x\in \mathrm{\Omega },u(x)=0,& x\in \mathrm{\partial }\mathrm{\Omega },\end{array}$$ under the conditions $$\frac{\alpha }{(1+\mathrm{\mid }s\mathrm{\mid }{)}^{\theta }{\ln }^{\theta }(e+\mathrm{\mid }s\mathrm{\mid })}\le a(x,s)\le \beta ,0<\alpha \le \beta <\mathrm{\infty },\theta \ge 0,$$ we obtain some regularity properties of its weak solutions.