[Paper Review] On finite Morse index solutions to the quadharmonic Lane-Emden equation

Senping Luo, Juncheng Wei+1 more|arXiv (Cornell University)|Sep 5, 2016

Nonlinear Partial Differential Equations16 references3 citations

TL;DR

This paper establishes the Joseph-Lundgren exponent for the quadharmonic Lane-Emden equation $\Delta^4 u = |u|^{p-1}u$ in $\mathbb{R}^n$, derives a monotonicity formula, and classifies finite Morse index solutions, proving that such solutions vanish identically when $p < p_c(n)$, with $p_c(n)$ explicitly computed for $n \geq 18$ and $p_c(n) = \infty$ for $n \leq 17$. The results extend Liouville-type theorems to higher-order elliptic equations.

ABSTRACT

In this paper, we compute the Joseph-Lundgren exponent for the quadharmonic Lane-Emden equation, derive a monotonicity formula and classify the finite Morse index solution to the following quadharmonic Lane-Emden equation: oindent \begin{equation} onumber Δ^4 u=|u|^{p-1}u\;\;\;\;\hbox{in}\;\;\;\;\; \R^n. \end{equation} As a byproduct, we also get a monotonicity formula for the quadharmonic maps $ Δ^4 u=0$.

Motivation & Objective

• To compute the Joseph-Lundgren exponent $p_c(n)$ for the quadharmonic Lane-Emden equation $\Delta^4 u = |u|^{p-1}u$ in $\mathbb{R}^n$.
• To derive a monotonicity formula for stable and finite Morse index solutions of the equation.
• To classify all finite Morse index solutions and prove Liouville-type theorems, showing that such solutions must be trivial ($u \equiv 0$) under certain $p$-dependent conditions.
• To extend existing Liouville-type results from second-order to fourth-order semilinear elliptic equations.

Proposed method

• Derives a monotonicity formula for solutions of $\Delta^4 u = |u|^{p-1}u$ using a weighted energy functional $E(r,0,u)$ involving $|\Delta^2 u|^2$ and $|u|^{p+1}$.
• Introduces a scaling argument via the blowing-down sequence $u^\lambda(x) = \lambda^{8/(p-1)} u(\lambda x)$ to analyze asymptotic behavior at infinity.
• Applies derivative estimates and decay bounds to control error terms in the monotonicity formula, showing $E(r,0,u)$ is bounded and eventually vanishes.
• Uses the homogeneity of the limit solution $u^\infty$ obtained from the scaling limit to deduce $u^\infty \equiv 0$ when $p < p_c(n)$, implying $u \equiv 0$.
• Employs a Pohozaev-type identity and integration by parts on annular regions to derive the energy identity $\int |\Delta^2 u|^2 = \int |u|^{p+1}$ for $p = \frac{n+8}{n-8}$.
• Establishes the explicit form of $p_c(n)$ via a complex algebraic expression involving nested radicals and high-degree polynomials in $n$, valid for $n \geq 18$.

Experimental results

Research questions

• RQ1 What is the precise value of the Joseph-Lundgren exponent $p_c(n)$ for the quadharmonic Lane-Emden equation in $\mathbb{R}^n$?
• RQ2 Under what conditions on $p$ do finite Morse index solutions to $\Delta^4 u = |u|^{p-1}u$ vanish identically?
• RQ3 Can a monotonicity formula be established for solutions of the quadharmonic equation, and how does it control the behavior of solutions at infinity?
• RQ4 What is the role of the critical exponent $p = \frac{n+8}{n-8}$ in the classification of solutions with finite energy?
• RQ5 How does the scaling limit $u^\lambda$ behave, and what does it reveal about the structure of solutions?

Key findings

• For $n \leq 17$, the Joseph-Lundgren exponent is $p_c(n) = \infty$, implying all finite Morse index solutions vanish when $p < \infty$, which holds for all $p > 1$.
• For $n \geq 18$, the Joseph-Lundgren exponent is $p_c(n) = \frac{n+6-2d(n)}{n-10-2d(n)}$, where $d(n)$ is a complex expression involving radicals and polynomials in $n$, and $p_c(n) \to \infty$ as $n \to \infty$.
• Finite Morse index solutions to $\Delta^4 u = |u|^{p-1}u$ vanish identically if $1 < p < \frac{n+8}{n-8}$ or $\frac{n+8}{n-8} < p < p_c(n)$.
• At the critical exponent $p = \frac{n+8}{n-8}$, finite Morse index solutions have finite energy: $\int_{\mathbb{R}^n} |\Delta^2 u|^2 = \int_{\mathbb{R}^n} |u|^{p+1} < \infty$.
• The monotonicity formula implies that $E(r,0,u) \to 0$ as $r \to \infty$ and $r \to 0$, leading to $E(r,0,u) \equiv 0$, which forces $u$ to be homogeneous and hence $u \equiv 0$ when $p < p_c(n)$.
• The scaling limit $u^\lambda$ converges to a homogeneous solution $u^\infty$, and if $p < p_c(n)$, then $u^\infty \equiv 0$, implying $|x|^{8/(p-1)}|u(x)| \to 0$ as $|x| \to \infty$.