The integer part of nonlinear forms with prime variables

In this paper, we discuss problems that integer part of nonlinear forms with prime variables represent primes infinitely. We prove that under suitable conditions there exist infinitely many primes $p_j,p$ such that $[{\lambda }_1{p}_1^2+{\lambda }_2{p}_2^2+{\lambda }_3{p}_3^k]=p$ and $[{\lambda }_1{p}_1^3+\cdots +{\lambda }_4{p}_4^3+{\lambda }_5{p}_5^k]=p$ with $k\ge 2$ and $k\ge 3$ respectively, which improve the author's earlier results.