Classification of Solutions to Several Semi-linear Polyharmonic Equations and Fractional Equations

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