[Paper Review] Coron problem for nonlocal equations invloving Choquard nonlinearity

Divya Goel, K. Sreenadh+1 more|arXiv (Cornell University)|Apr 22, 2018

Nonlinear Partial Differential Equations40 references6 citations

TL;DR

This paper establishes the existence of a positive high-energy solution to a nonlocal Choquard equation with critical Hardy–Littlewood–Sobolev exponent in an annular-type domain when the inner hole is sufficiently small. Using variational methods, Pohozaev-type identities, and a global compactness lemma, the authors prove the existence of a critical point via the deformation lemma and contradiction, leveraging topological constraints of the domain.

ABSTRACT

We study the problem \[ -\De u = \left(\int_{\Om}\frac{|u(y)|^{2^*_μ}}{|x-y|^μ}dy ight)|u|^{2^*_μ-2}u, \; ext{in}\; \Om,\quad u = 0 \; ext{ on } \pa \Om , \] where $\Om$ is a smooth bounded domain in $\mathbb{R}^N( N\geq 3)$, $2^*_μ=\frac{2N-μ}{N-2}$. we prove the existence of a positive solution of the above problem in an annular type domain when the inner hole is sufficiently small.

Motivation & Objective

• To address the open problem of existence and multiplicity of solutions for nonlocal equations with Choquard nonlinearity in non-contractible domains.
• To extend Coron's classical result on critical elliptic equations to the nonlocal setting involving Riesz potentials and the Hardy–Littlewood–Sobolev inequality.
• To establish a non-existence result via Pohozaev identity in the half-space and a global compactness lemma for the Choquard equation in bounded domains.
• To prove the existence of a positive high-energy solution in an annular domain when the inner hole is small, using topological and variational arguments.

Proposed method

• Formulate the Choquard equation with critical exponent $2^*_\mu = (2N - \mu)/(N - 2)$ and convolution-type nonlinearity involving Riesz potential.
• Use the Pohozaev identity to prove non-existence of solutions in $\mathbb{R}^N_+$, establishing a necessary condition for existence.
• Prove a global compactness lemma for the Choquard equation in bounded domains, essential for handling lack of $C^2$ regularity when $\mu > \min\{4, N\}$.
• Apply the deformation lemma and contradiction argument via a continuous deformation map $D: \Sigma \times [0,1] \to \overline{\Omega}$ to detect a critical point.
• Use concentration-compactness principles to analyze weak limits of sequences and show concentration of energy at a point in the domain.
• Leverage topological invariance: if $\Sigma$ is a sphere and $D$ is a contraction, a contradiction arises, implying existence of a critical point.

Experimental results

Research questions

• RQ1 Does the Choquard equation with critical Hardy–Littlewood–Sobolev exponent admit a positive high-energy solution in a non-contractible domain?
• RQ2 Can the Coron-type problem for nonlocal equations with convolution nonlinearity be resolved using variational and topological methods?
• RQ3 What role does the size of the inner hole in an annular domain play in the existence of high-energy solutions?
• RQ4 How does the lack of $C^2$ regularity in the energy functional affect the existence theory for Choquard equations?
• RQ5 Can the Palais–Smale condition be established for the associated functional in the relevant energy range?

Key findings

• A positive high-energy solution exists for the Choquard equation in an annular domain when the inner hole is sufficiently small.
• The functional associated with the problem satisfies the Palais–Smale condition in the range $\left(S_{H,L}, 2^{\frac{N-\mu+2}{2N-\mu}} S_{H,L}\right)$, enabling critical point theory.
• The best constant $S_{H,L}$ in the Hardy–Littlewood–Sobolev inequality is attained in the limit, and the concentration-compactness principle confirms energy concentration at a point.
• The contradiction argument based on the deformation lemma and the map $D$ shows that the domain's topology (non-contractibility) forces the existence of a critical point.
• The solution $u$ belongs to $C^2(\overline{\Omega}) \cap L^\infty(\Omega)$, and by the maximum principle, it is strictly positive in $\Omega$.
• The energy level of the solution is $\frac{N - \mu + 2}{2(2N - \mu)} c^{\frac{2N - \mu}{N - \mu + 2}}$, where $c$ is the limit of the sequence norm.