A Generalization of the Stampacchia Lemma and Applications

We present a generalization of Stampacchia Lemma and give applications to regularity property of weak and entropy solutions of degenerate elliptic equations of the form $$\begin{cases} -\text{div} (a(x,u(x)) Du (x)) =f(x), & \text { in } \Omega, \\[1mm] u(x)=0, & \text { on } \partial \Omega, \end{cases}$$ where $\displaystyle\frac {\alpha}{(1+|u|) ^\theta} \le a(x,s)\le \beta$ with $0<\alpha \le \beta <\infty$ and $0\le \theta <1$. The method to derive regularity seems to be simpler than the classical ones.