[Paper Review] Uniform a priori estimates for positive solutions of higher order Lane-Emden equations in $\mathbb{R}^n$

Wei Dai, Thomas Duyckaerts|arXiv (Cornell University)|May 24, 2019

Nonlinear Partial Differential Equations16 references4 citations

TL;DR

This paper establishes uniform a priori estimates for positive solutions of higher-order Lane-Emden equations with Navier boundary conditions in star-shaped or strictly convex bounded domains in $\mathbb{R}^n$, extending prior results from the second-order case ($m=1$) to general $m \geq 2$. Using generalized Pohozaev-type identities and scaling arguments, the authors prove that the $L^\infty$-norm of solutions remains uniformly bounded as $p \to p_c^-$, establishing a critical threshold for existence and blow-up behavior.

ABSTRACT

In this paper, we study the existence of uniform a priori estimates for positive solutions to Navier problems of higher order Lane-Emden equations \begin{equation*} (-Δ)^{m}u(x)=u^{p}(x), \qquad \,\, x\inΩ\end{equation*} for all large exponents $p$, where $Ω\subset\mathbb{R}^{n}$ is a star-shaped or strictly convex bounded domain with $C^{2m-2}$ boundary, $n\geq4$ and $2\leq m\leq\frac{n}{2}$. Our results extend those of previous authors for second order $m=1$ to general higher order cases $m\geq2$.

Motivation & Objective

• To extend uniform a priori estimates for positive solutions of higher-order Lane-Emden equations beyond the second-order case ($m=1$) to general $m \geq 2$.
• To establish uniform boundedness of the $L^\infty$-norm of positive solutions as $p$ approaches the critical exponent $p_c = \frac{n+2m}{n-2m}$ from below.
• To prove that no positive solutions exist for $p \geq p_c$ in star-shaped domains, generalizing the classical nonexistence result for $m=1$.
• To develop and apply generalized Pohozaev-type variational identities for higher-order Navier problems with $C^{2m-2}$ boundaries.

Proposed method

• Derivation of a generalized Pohozaev-type identity for higher-order Navier problems involving $(-\Delta)^m u = f(u)$ with Navier boundary conditions.
• Application of the Pohozaev identity to prove Liouville-type theorems for nonnegative solutions under star-shaped domain assumptions.
• Use of scaling techniques and Green's representation formula to estimate pointwise values of derivatives $(-\Delta)^k u(x_k)$ at concentration points.
• Employment of inhomogeneous Harnack inequalities to control the decay of $u(x)$ near the maximum point $x=0$, linking $L^\infty$-norm to the behavior of $u$ in small balls.
• Construction of a lower bound for $\int_{B_r(0)} \ln(1/|x|) u^p(x) dx$ using the maximum value $M = \|u\|_{L^\infty}$ and asymptotic analysis as $p \to p_c^-$.
• Combining estimates from the Pohozaev identity and Green's representation to derive a contradiction unless $M$ is uniformly bounded, leading to the key a priori estimate.

Experimental results

Research questions

• RQ1 Can uniform a priori estimates for positive solutions of higher-order Lane-Emden equations be established for $m \geq 2$ in the same way as for $m=1$?
• RQ2 What is the behavior of the $L^\infty$-norm of positive solutions as $p \to p_c^-$ for $m \geq 2$ in star-shaped or strictly convex domains?
• RQ3 Does the nonexistence of positive solutions for $p \geq p_c$ in star-shaped domains extend from $m=1$ to higher-order cases?
• RQ4 How can Pohozaev-type identities be generalized to handle Navier boundary conditions for $m \geq 2$?
• RQ5 Can scaling and Harnack-type estimates be used to derive uniform bounds on the $L^\infty$-norm independent of $p$?

Key findings

• The $L^\infty$-norm of positive solutions to the higher-order Lane-Emden equation remains uniformly bounded as $p \to p_c^-$, independent of $p$, for $m \geq 2$ and $n \geq 4$, $m < n/2$.
• A uniform a priori estimate $\|u\|_{L^\infty(\overline{\Omega})} \leq e^C$ holds for all $p \in (1, p_c)$, with $C$ independent of $p$, when $\Omega$ is star-shaped or strictly convex with $C^{2m-2}$ boundary.
• The generalized Pohozaev identity for higher-order Navier problems involves a sum of boundary integrals over iterated Laplacian derivatives, capturing the structure of the solution's growth and decay.
• For $p$ close to $p_c$, the solution concentrates near a point, and the maximum value $M = \|u\|_{L^\infty}$ satisfies $M \leq e^C$, implying no blow-up as $p \to p_c^-$.
• The method relies on a delicate balance between the Green's representation of $(-\Delta)^k u$ and logarithmic integral estimates, leading to a contradiction if $M$ is too large.
• The result confirms that the critical exponent $p_c = \frac{n+2m}{n-2m}$ acts as a sharp threshold for existence and uniform boundedness of positive solutions in higher-order settings.