We consider the following semi-linear equations (-Δ)pu=u+γinRn,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} (-\Delta )^pu=u^\gamma _+ ~~ \text{ in } {{\mathbb {R}}^n}, \end{aligned}$$\end{document}where γ∈(1,n+2pn-2p)\documentclass[12pt]{minimal}
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\begin{document}$$\gamma \in (1,\frac{n+2p}{n-2p})$$\end{document}, n>2p>0\documentclass[12pt]{minimal}
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\begin{document}$$n>2p>0$$\end{document}, u+=max{u,0}\documentclass[12pt]{minimal}
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\begin{document}$$u_+=\max \{u,0\}$$\end{document}, and 2≤p∈N\documentclass[12pt]{minimal}
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\begin{document}$$2\le p\in {\mathbb {N}}$$\end{document} or p∈(0,1)\documentclass[12pt]{minimal}
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\begin{document}$$p\in (0,1)$$\end{document}. Subject to the integral constraint u+γ∈L1(Rn),\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} u_+^\gamma \in L^1({\mathbb {R}}^n), \end{aligned}$$\end{document}we obtain the classification of solutions to the above polyharmonic equation for any γ<n+2pn-2p\documentclass[12pt]{minimal}
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\begin{document}$$\gamma <\frac{n+2p}{n-2p}$$\end{document} and γ≤nn-2p\documentclass[12pt]{minimal}
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\begin{document}$$\gamma \le \frac{n}{n-2p}$$\end{document}, according to the two different assumptions: Δu(x)→0\documentclass[12pt]{minimal}
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\begin{document}$$\Delta u(x)\rightarrow 0$$\end{document} and u(x)=o(|x|2)\documentclass[12pt]{minimal}
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\begin{document}$$u(x)=\text{ o }(|x|^2)$$\end{document} at infinity, respectively. Under the other integral constraint u+q∈L1(Rn),q=n(γ-1)2p,γ<n+2pn-2p,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} u_+^q\in L^1({\mathbb {R}}^n), \quad q=\frac{n(\gamma -1)}{2p},\quad \gamma <\frac{n+2p}{n-2p}, \end{aligned}$$\end{document}which is scaling invariant, the classification of solutions with the decay assumption Δu(x)→0\documentclass[12pt]{minimal}
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\begin{document}$$\Delta u(x)\rightarrow 0$$\end{document} at infinity is established for any integer p≥2\documentclass[12pt]{minimal}
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\begin{document}$$p\ge 2$$\end{document}, and the classification of solutions with the growth assumption u(x)=o(|x|2)\documentclass[12pt]{minimal}
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\begin{document}$$u(x)=\text{ o }(|x|^2)$$\end{document} at infinity is proved for integers p=2,3\documentclass[12pt]{minimal}
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\begin{document}$$p=2, 3$$\end{document} as well. In the fractional equation case, namely p∈(0,1)\documentclass[12pt]{minimal}
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\begin{document}$$p\in (0,1)$$\end{document}, under either of the above two integral constraints, we also complete the classification of solutions with certain growth assumption at infinity.
