[Paper Review] Existence and multiplicity results for Brezis-Nirenberg type fractional Choquard equation

Tuhina Mukherjee, K. Sreenadh|arXiv (Cornell University)|May 22, 2016

Nonlinear Partial Differential Equations21 references6 citations

TL;DR

This paper investigates the existence and multiplicity of solutions to a fractional Choquard equation of Brezis-Nirenberg type involving the fractional Laplacian and a Hardy-Littlewood-Sobolev critical nonlinearity. Using variational methods, the authors establish the existence of nontrivial solutions for certain parameter regimes, particularly when the parameter λ is below a critical threshold.

ABSTRACT

In this article, we study the Brezis-Nirenberg type problem of nonlinear Choquard equation involving a fractional Laplacian \[ (-\De)^s u = \left( \int_{\Om}\frac{|u|^{2^*_{\mu,s}}}{|x-y|^{\mu}}\mathrm{d}y ight)|u|^{2^*_{\mu,s}-2}u +\la u \; ext{in } \Om,\] where $\Om $ is a bounded domain in $\mathbb R^n$ with $C^{1,1}$ boundary, $\la $ is a real parameter, $s \in (0,1)$, $n >2s$ and $2^*_{\mu,s}= (2n-\mu)/(n-2s)$ is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality. We obtain some existence and multiplicity results for solution of the above equation using variational methods.

Motivation & Objective

• To study the Brezis-Nirenberg type problem for a nonlinear Choquard equation involving the fractional Laplacian in a bounded domain with C^{1,1} boundary.
• To analyze the role of the parameter λ in determining the existence and multiplicity of solutions.
• To address the critical exponent $2^*_{ ho,s} = (2n - \mu)/(n - 2s)$ arising from the Hardy-Littlewood-Sobolev inequality.
• To extend variational methods to the fractional nonlocal setting with a singular, nonlocal integral nonlinearity.
• To establish conditions under which nontrivial solutions exist and are multiple.

Proposed method

• Employing variational methods, particularly the critical point theory, to analyze weak solutions of the equation.
• Working in the Sobolev space $H^s_0(\Omega)$, which is the natural space for the fractional Laplacian with zero boundary conditions.
• Using the Hardy-Littlewood-Sobolev inequality to justify the well-posedness of the nonlocal term $\int_{\Omega} \frac{|u|^{2^*_{\mu,s}}}{|x-y|^\mu} dy$.
• Applying concentration-compactness principles and compact embeddings to overcome lack of compactness in the critical exponent case.
• Analyzing the energy functional associated with the equation and proving the existence of critical points via the mountain pass lemma or other variational tools.
• Establishing the existence of at least one nontrivial solution when $\lambda$ is below a certain threshold, and multiple solutions under additional geometric or symmetry conditions.

Experimental results

Research questions

• RQ1 Under what conditions does the fractional Choquard equation with a Brezis-Nirenberg type nonlinearity admit at least one nontrivial solution?
• RQ2 How does the parameter $\lambda$ influence the existence and multiplicity of solutions in the critical exponent regime?
• RQ3 What role does the fractional order $s \in (0,1)$ play in the solvability of the equation?
• RQ4 Can variational methods be successfully applied to this nonlocal, nonlinearity involving the Hardy-Littlewood-Sobolev integral?
• RQ5 Are there multiple solutions when $\lambda$ lies in a certain range below the first eigenvalue of the fractional Laplacian?

Key findings

• Nontrivial weak solutions exist for the fractional Choquard equation when the parameter $\lambda$ is below a certain critical threshold.
• The critical exponent $2^*_{\mu,s} = (2n - \mu)/(n - 2s)$ arises naturally from the Hardy-Littlewood-Sobolev inequality and governs the growth of the nonlinearity.
• The use of variational methods allows the authors to prove existence results despite the nonlocal nature of the Choquard term.
• The bounded domain $\Omega \subset \mathbb{R}^n$ with $C^{1,1}$ boundary ensures sufficient regularity for the analysis.
• The solution set exhibits multiplicity under suitable conditions on $\lambda$, particularly when $\lambda$ is below the first eigenvalue of $(-\Delta)^s$.
• The results extend classical Brezis-Nirenberg results to the fractional and nonlocal setting with a convolution-type nonlinearity.