  Let $\mathbb{F}_q$ be a finite field of order $q$, and $P$ be the paraboloid
in $\mathbb{F}_q^3$ defined by the equation $z=x^2+y^2$. A tuple $(a, b, c,
d)\in P^4$ is called a non-trivial energy tuple if $a+b=c+d$ and $a, b, c, d$
are distinct. For $X\subset P$, let $\mathcal{E}^+(X)$ be the number of
non-trivial energy tuples in $X$. It was proved recently by Lewko (2020) that
$\mathcal{E}^+(X)\ll |X|^{\frac{99}{41}}$ for $|X|\ll q^{\frac{26}{21}}$. The
main purposes of this paper are to prove lower bounds of $\mathcal{E}^+(X)$ and
to study related questions by using combinatorial arguments and a weak
hypergraph regularity lemma developed recently by Lyall and Magyar (2020).
