[Paper Review] On Isolated Singularities of Fractional Semi-Linear Elliptic Equations

Hui Yang, Wenming Zou|arXiv (Cornell University)|Apr 3, 2018

Nonlinear Partial Differential Equations3 citations

TL;DR

This paper investigates the local behavior of nonnegative solutions to the fractional semi-linear elliptic equation $(-\Delta)^\sigma u = u^p$ with an isolated singularity at the origin, where $\sigma \in (0,1)$ and $\frac{n}{n-2\sigma} < p < \frac{n+2\sigma}{n-2\sigma}$. Using the Caffarelli-Silvestre extension and a blow-up method combined with a Liouville-type theorem, the authors establish a sharp upper bound and prove that either the singularity is removable or the solution behaves like $|x|^{-2\sigma/(p-1)}$ near zero, extending classical results for the Laplacian ($\sigma=1$).

ABSTRACT

In this paper, we study the local behavior of nonnegative solutions of fractional semi-linear equations $(-Δ)^σu = u^p$ with an isolated singularity, where $\sg \in (0, 1)$ and $\frac{n}{n-2\sg} &lt; p &lt; \frac{n+2\sg}{n-2\sg}$. We first use blow up method and a Liouville type theorem to derive an upper bound. Then we establish a monotonicity formula and a sufficient condition for removable singularity to give a classification of the isolated singularities. When $\sg=1$, this classification result has been proved by Gidas and Spruck (Comm. Pure Appl. Math. 34: 525-598, 1981).

Motivation & Objective

• To classify the nature of isolated singularities in nonnegative solutions of $(-\Delta)^\sigma u = u^p$ in $B_1 \setminus \{0\}$, where $\sigma \in (0,1)$ and $p$ is in the subcritical range.
• To extend the classical classification result of Gidas and Spruck (1981) for the Laplacian ($\sigma=1$) to the fractional Laplacian case.
• To establish sharp pointwise estimates for the decay rate of singular solutions near the origin using the extension method and monotonicity formulas.
• To prove that either the singularity is removable (i.e., $u$ extends continuously to the origin), or $u(x) \sim |x|^{-2\sigma/(p-1)}$ as $x \to 0$.
• To overcome the lack of Pohozaev identity in the fractional setting by constructing a suitable barrier function and testing against a carefully chosen cut-off.

Proposed method

• Employ the Caffarelli-Silvestre extension to convert the fractional equation $(-\Delta)^\sigma u = u^p$ in $\mathbb{R}^n \setminus \{0\}$ into a degenerate elliptic problem in the upper half-space $\mathbb{R}^{n+1}_+$.
• Use the blow-up method and a Liouville-type theorem to derive an upper bound of the form $u(x) \leq c |x|^{-2\sigma/(p-1)}$.
• Construct a barrier function $\Phi = |X|^{-\tau} - \delta t^{2\sigma} |X|^{-(\tau+2\sigma)}$ with $\tau = \frac{n-2\sigma}{p-1}(p - \frac{n}{n-2\sigma})$ to test the behavior of solutions near the origin.
• Apply a weighted divergence theorem and test the solution $U$ against the barrier $\zeta \Phi$, where $\zeta$ is a radial cut-off function, to derive integral estimates.
• Use the Harnack inequality to control the growth of $U$ in the extended space and derive uniform bounds independent of $\epsilon$.
• Leverage the decay of $u^{p-1}$ as $|x| \to 0$ under the assumption $\liminf_{x \to 0} |x|^{2\sigma/(p-1)} u(x) = 0$ to prove integrability and conclude removability of the singularity.

Experimental results

Research questions

• RQ1 What is the precise asymptotic behavior of nonnegative solutions to $(-\Delta)^\sigma u = u^p$ with an isolated singularity at the origin for $\sigma \in (0,1)$ and $\frac{n}{n-2\sigma} < p < \frac{n+2\sigma}{n-2\sigma}$?
• RQ2 Under what conditions is the isolated singularity removable, i.e., when can the solution be extended continuously to the origin?
• RQ3 How does the solution decay near the origin when the singularity is non-removable?
• RQ4 Can the classical classification result of Gidas and Spruck for $\sigma=1$ be extended to the fractional case $\sigma \in (0,1)$?
• RQ5 How can one overcome the absence of the Pohozaev identity in the fractional setting to derive pointwise estimates?

Key findings

• The solution $u(x)$ satisfies the upper bound $u(x) \leq c_2 |x|^{-2\sigma/(p-1)}$ for some $c_2 > 0$ near the origin.
• If the singularity is non-removable, then $u(x)$ satisfies the sharp lower bound $c_1 |x|^{-2\sigma/(p-1)} \leq u(x)$ for some $c_1 > 0$, so $u(x) \sim |x|^{-2\sigma/(p-1)}$ as $x \to 0$.
• The extended solution $U(X)$ in $\mathbb{R}^{n+1}_+$ satisfies $C_1 |X|^{-2\sigma/(p-1)} \leq U(X) \leq C_2 |X|^{-2\sigma/(p-1)}$ near the origin due to the Harnack inequality.
• When $\liminf_{x \to 0} |x|^{2\sigma/(p-1)} u(x) = 0$, the singularity is removable, meaning $u$ extends continuously to the origin.
• The proof relies on constructing a barrier function $\Phi$ and testing against a cut-off function $\zeta$, leading to uniform $L^1$ bounds on $u |x|^{-(n - 2\sigma/(p-1))}$, which implies integrability and removability.
• The result generalizes the classical $\sigma=1$ case of Gidas and Spruck (1981) to the fractional setting, establishing a complete classification of isolated singularities for the subcritical range.