On the exceptional set for Diophantine inequality with unlike powers of primes

Let λ2, λ3, λ4, λ5 be nonzero real numbers, not all negative. Let V\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak{V}$$\end{document} be a well-spaced sequence. Assume that λ2/λ3 is irrational and algebraic, and δ > 0. Let EV,N,δ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E\left(\mathfrak{V},N,\delta \right)$$\end{document} be the number of υ∈V\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\upsilon \in \mathfrak{V}$$\end{document} with υ≤N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\upsilon \le N$$\end{document} such that the Diophantine inequality λ2p22+λ3p33+λ4p44+λ5p55-υ<υ-δ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left|{\lambda }_{2}{p}_{2}^{2}+{\lambda }_{3}{p}_{3}^{3}+{\lambda }_{4}{p}_{4}^{4}+{\lambda }_{5}{p}_{5}^{5}-\upsilon \right|<{\upsilon }^{-\delta }$$\end{document} has no solution in primes p2, p3, p4, p5. In this paper, we prove that for any ε>0,EV,N,δ≪N1-19/378+2δ+ε,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon >0,E\left(\mathfrak{V},N,\delta \right)\ll {N}^{1-19/378+2\delta +\varepsilon },$$\end{document} which refines the previous result.