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

Abstract

We consider the following semi-linear equations

$$\begin{aligned} (-\Delta )^pu=u^\gamma _+ ~~ \text{ in } {{\mathbb {R}}^n}, \end{aligned}$$

where $\gamma \in (1,\frac{n+2p}{n-2p})$, $n>2p>0$, $u_+=\max \{u,0\}$, and $2\le p\in {\mathbb {N}}$ or $p\in (0,1)$. Subject to the integral constraint

$$\begin{aligned} u_+^\gamma \in L^1({\mathbb {R}}^n), \end{aligned}$$

we obtain the classification of solutions to the above polyharmonic equation for any $\gamma <\frac{n+2p}{n-2p}$ and $\gamma \le \frac{n}{n-2p}$, according to the two different assumptions: $\Delta u(x)\rightarrow 0$ and $u(x)=\text{ o }(|x|^2)$ at infinity, respectively. Under the other integral constraint

$$\begin{aligned} u_+^q\in L^1({\mathbb {R}}^n), \quad q=\frac{n(\gamma -1)}{2p},\quad \gamma <\frac{n+2p}{n-2p}, \end{aligned}$$

which is scaling invariant, the classification of solutions with the decay assumption $\Delta u(x)\rightarrow 0$ at infinity is established for any integer $p\ge 2$, and the classification of solutions with the growth assumption $u(x)=\text{ o }(|x|^2)$ at infinity is proved for integers $p=2, 3$ as well. In the fractional equation case, namely $p\in (0,1)$, under either of the above two integral constraints, we also complete the classification of solutions with certain growth assumption at infinity.