[Paper Review] Non degeneracy of the bubble in the critical case for non local equations

Juan Dávila, Manuel del Pino+1 more|arXiv (Cornell University)|Feb 1, 2013

Nonlinear Partial Differential Equations10 references8 citations

TL;DR

This paper establishes the nondegeneracy of the extremal functions for the fractional Sobolev inequality in the critical case, proving that all bounded solutions to the linearized equation around the bubble solution are spanned by the generators of scaling and translation invariance. The proof uses stereographic projection to transform the nonlocal equation into an integral equation on the sphere, where spectral theory of integral operators on $\mathbb{S}^N$ confirms that only the first-order spherical harmonics solve the linearized problem.

ABSTRACT

We prove the nondegeneracy of the extremals of the fractional Sobolev inequality as solutions of a critical semilinear nonlocal equation involving the fractional Laplacian.

Motivation & Objective

• To establish the linear nondegeneracy of the extremal functions for the critical fractional Sobolev inequality.
• To show that the only bounded solutions to the linearized equation around the fractional bubble are generated by scaling and translation invariance.
• To extend the classical nondegeneracy result for the local case ($s=1$) to the nonlocal fractional setting ($0<s<1$).
• To provide a rigorous spectral-theoretic justification of the uniqueness of the kernel of the linearized operator in the nonlocal setting.

Proposed method

• Transform the nonlocal equation via stereographic projection to map $\mathbb{R}^N$ to the sphere $\mathbb{S}^N$, preserving integral structure.
• Use the Jacobian of the stereographic projection to rewrite the integral equation in terms of functions on $\mathbb{S}^N$.
• Define a transformed function $h = J^{2s/N} \circ S^{-1} \tilde{\phi}$ to convert the original equation into a compact integral operator on $L^2(\mathbb{S}^N)$.
• Apply spectral theory of integral operators with Riesz-type kernels to identify eigenvalues and eigenspaces on $\mathbb{S}^N$.
• Use the known decomposition of $L^2(\mathbb{S}^N)$ into spaces of spherical harmonics $H_l$ to analyze the solution space.
• Show that the only bounded solutions correspond to the $l=1$ spherical harmonics, which correspond to the physical symmetries of the problem.

Experimental results

Research questions

• RQ1 Are the extremal functions of the fractional Sobolev inequality nondegenerate in the critical case?
• RQ2 What is the structure of the kernel of the linearized operator around the fractional bubble solution?
• RQ3 Can the nondegeneracy property in the local case ($s=1$) be extended to the nonlocal fractional case ($0<s<1$)?
• RQ4 Do the only bounded solutions of the linearized equation arise from the symmetries of scaling and translation?
• RQ5 What spectral properties of the integral operator on the sphere determine the solution space of the linearized problem?

Key findings

• The extremal functions $w_{\mu,\xi}(x) = \alpha_{N,s} \left( \frac{\mu}{\mu^2 + |x - \xi|^2} \right)^{\frac{N-2s}{2}}$ are nondegenerate in the sense that their linearized kernel is exactly the span of the generators of scaling and translation invariance.
• All bounded solutions to the linearized equation $(-\Delta)^s \phi = p w^{p-1} \phi$ with $p = \frac{N+2s}{N-2s}$ are linear combinations of $\frac{N-2s}{2}w + x \cdot \nabla w$ and $\partial_{x_i} w$ for $1 \leq i \leq N$.
• The transformed function $h$ on $\mathbb{S}^N$ satisfies an integral equation with a compact self-adjoint operator, whose only nonzero eigenvalues correspond to the $l=1$ spherical harmonics.
• The eigenvalue $a$ in the transformed equation matches the eigenvalue $e_1$ of the integral operator, confirming that only $H_1$-functions solve the equation.
• The boundedness of $\phi$ implies decay $|\phi(x)| \leq C / |x|^{N-2s}$ for large $|x|$, which ensures the continuity and boundedness of $h$ on $\mathbb{S}^N$.
• The solution space is exactly $N+1$-dimensional, matching the dimension of $H_1$, and is generated by the symmetries of the original equation.