  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 $$ {\cal J}
(u;\Omega) = \int_{\Omega}\ f(x,Du{(x)})dx. $$ We consider a minimizer $u:
\Omega \subset \mathbb R^n \to \mathbb R $ among all functions with a fixed
boundary value $u_{\ast }$
  on $\partial \Omega$. 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 $$ \begin{cases} -\mbox
{div} \left( a(x, u(x)) D u(x) \right) = f(x), & x \in \Omega,
  u(x) = 0, & x \in {\partial \Omega}, \end{cases} $$ under the conditions $$
\frac {\alpha }{(1+|s|) ^\theta \ln ^\theta (e+|s|)} \le a (x,s) \le \beta, \ \
\ 0<\alpha \le \beta <\infty, \ \theta \ge 0, $$ we obtain some regularity
properties of its weak solutions.
