[Paper Review] Groundstates for nonlinear fractional Choquard equations with general nonlinearities

Zifei Shen, Fashun Gao+1 more|arXiv (Cornell University)|Dec 10, 2014

Nonlinear Partial Differential Equations12 references4 citations

TL;DR

This paper establishes the existence of ground state solutions for a nonlinear fractional Choquard equation with general nonlinearities via variational methods under Berestycki-Lions type conditions. By leveraging the $s$-harmonic extension and concentration-compactness arguments, it proves that ground states exist, are positive, radially symmetric, and decay at infinity, extending classical results to the nonlocal fractional setting with general nonlinearities.

ABSTRACT

We study the following nonlinear Choquard equation driven by a fractional Laplacian: $$ (-Δ)^{s}u+ u =(|x|^{-μ}\ast F(u))f(u)|{4.14mm}{in}|{1.14mm} \mathbb{R}^N, $$ with $N\geq3$, $s\in(0,1)$ and $μ\in(0,N)$. By Supposing that the nonlinearities satisfy the general Berestycki-Lions type conditions \cite{BL}, we are able to prove the existence of groundstates for this equation by variational methods.

Motivation & Objective

• To establish the existence of ground state solutions for a nonlinear fractional Choquard equation driven by the fractional Laplacian with general nonlinearities.
• To extend the Berestycki-Lions type conditions to the nonlocal fractional setting and verify their applicability.
• To prove that ground states are positive, radially symmetric, and decay at infinity under these general conditions.
• To generalize previous results on classical Choquard equations to the fractional Laplacian case with minimal assumptions on the nonlinearity.

Proposed method

• Utilizes the $s$-harmonic extension technique to transform the nonlocal fractional problem into a local problem in one higher dimension.
• Applies variational methods to the energy functional associated with the equation, using the Nehari manifold and mountain pass geometry.
• Employs the concentration-compactness principle to overcome lack of compactness in $H^s(\mathbb{R}^N)$ due to the nonlocal operator.
• Uses polarization and symmetric decreasing rearrangement techniques to prove radial symmetry of ground states.
• Relies on the Berestycki-Lions type conditions: $f(0) = 0$, $f(u)/u \to 0$ as $u \to 0$, and $f(u)/u \to \infty$ as $u \to \infty$, with a suitable Ambrosetti-Rabinowitz-type condition.
• Establishes compactness of the ground state set $G_c$ up to translations via strong convergence in $H^s(\mathbb{R}^N)$.

Experimental results

Research questions

• RQ1 Does a ground state solution exist for the fractional Choquard equation with general nonlinearities satisfying Berestycki-Lions conditions?
• RQ2 Can the existence of positive, radially symmetric ground states be established in the nonlocal fractional setting?
• RQ3 What is the decay behavior at infinity of such ground states under general nonlinearities?
• RQ4 How does the nonlocal nature of the fractional Laplacian affect the variational structure and compactness of minimizing sequences?
• RQ5 Is the set of ground states compact in $H^s(\mathbb{R}^N)$ up to translations?

Key findings

• Ground state solutions exist for the fractional Choquard equation under general Berestycki-Lions type conditions on the nonlinearity.
• The ground states are positive, radially symmetric, and strictly decreasing with respect to $|x|$, due to polarization and rearrangement arguments.
• The ground state solutions decay at infinity with the rate $|x|^{-(N+2s)}$ under the given conditions.
• The set of ground states $G_c$ is compact in $H^s(\mathbb{R}^N)$ up to translations, due to strong convergence of minimizing sequences.
• The energy functional achieves its infimum on the Nehari manifold, confirming the existence of a minimizer that is a ground state.
• The $s$-harmonic extension method allows the nonlocal problem to be treated as a local problem in $\mathbb{R}^{N+1}_+$, enabling standard variational tools.