In this talk we are concerned with the existence of $L^p$-viscosity solutions of fully nonlinear, uniformly elliptic, second-order PDEs: $F(x, Du, D^2u) = f(x)$ in $Ω ⊂ R^n$, where $x → F(x, q, ξ)$ is only measurable, $q → F(x, q, ξ)$ may have quadratic growth, and $f ∈ L^p(Ω)$ under Dirichlet condition. The notion of $L^p$-viscosity solutions was introduced in a paper by Caffarelli–Crandall–Kocan–Swiech in 1996, which was motivated by Caffarelli’s regularity theory for viscosity solutions in 1989. In those works, it was assumed that $q → F(x, q, ξ)$ has linear growth. We note that there is a non-existence result of solutions by Nagumo when $F$ is quadratic in $q$. Hence, under certain hypotheses, we will get some a priori estimates ($L^∞$-bound and Holder continuity) to apply to two kinds of known existence results.