Roth-type Theorem for high-power system in Piatetski-Shapiro primes (II)

We consider the nonlinear system $c_1{p}_1^d+c_2{p}_2^d+\cdots +c_s{p}_s^d=0$ with $c_1,c_2,\dots ,c_s\in \mathbb{Z}$ being nonzero and satisfying $c_1+c_2+\cdots +c_s=0$. We show that for $s\ge 2\lfloor \frac{d^2}{2}\rfloor +1$ and $c\in (1,1+c(d,s))$, if the system has only $K$-trivial solutions in subset $\mathcal{A}$ of Piatetski-Shapiro primes up to $x$ and corresponding to $c$, then $\mathrm{\mid }\mathcal{A}\mathrm{\mid }\ll \frac{x^{\frac{1}{c}}}{\log x}$${(\log \log \log \log x)}^{\frac{2-s}{dc}+\epsilon }$.