[Paper Review] Existence and multiplicity of solutions for a class of Choquard equations with Hardy-Littlewood-Sobolev critical exponent

Fashun Gao, Minbo Yang|arXiv (Cornell University)|May 17, 2016

Nonlinear Partial Differential Equations31 references6 citations

TL;DR

This paper establishes the existence and multiplicity of solutions for a nonlinear Choquard equation with Hardy-Littlewood-Sobolev critical exponent in a bounded domain using variational methods. Under suitable conditions on the nonlinearity $ f(u) $, it proves the existence of at least one nontrivial solution and, under additional assumptions, multiple solutions via critical point theory and concentration-compactness arguments.

ABSTRACT

We consider the following nonlinear Choquard equation with Dirichlet boundary condition $$-\Delta u =\left(\int_{\Omega}\frac{|u|^{2_{\mu}^{\ast}}}{|x-y|^{\mu}}dy ight)|u|^{2_{\mu}^{\ast}-2}u+\lambda f(u)\hspace{4.14mm}\mbox{in}\hspace{1.14mm} \Omega, $$ where $\Omega$ is a smooth bounded domain of $\mathbb{R}^N$, $\lambda>0$, $N\geq3$, $0<\mu<N$ and $2_{\mu}^{\ast}$ is the critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality. Under suitable assumptions on different types of nonlinearities $f(u)$, we are able to prove some existence and multiplicity results for the equation by variational methods.

Motivation & Objective

• To investigate the existence and multiplicity of solutions for a Choquard equation with critical growth defined by the Hardy-Littlewood-Sobolev inequality.
• To analyze the impact of different types of nonlinearities $ f(u) $ on the solution structure.
• To extend variational methods to equations involving nonlocal interactions and critical exponents in bounded domains.
• To establish sufficient conditions under which multiple nontrivial solutions exist.

Proposed method

• Employing variational methods, particularly the critical point theory, to study weak solutions of the Choquard equation.
• Using the Hardy-Littlewood-Sobolev inequality to define the critical exponent $ 2_{ u}^{ullet} = \frac{2N - \mu}{N - 2} $, which governs the nonlocal term.
• Applying concentration-compactness principles to handle the lack of compactness due to the critical exponent and nonlocal integral term.
• Analyzing the energy functional associated with the equation to identify critical points corresponding to weak solutions.
• Imposing structural conditions on $ f(u) $, such as subcritical or asymptotic behavior, to ensure the Palais-Smale condition.
• Using comparison and embedding theorems in Sobolev spaces to control the nonlocal term and ensure boundedness of minimizing sequences.

Experimental results

Research questions

• RQ1 Under what conditions on $ f(u) $ does the Choquard equation with Hardy-Littlewood-Sobolev critical exponent admit at least one nontrivial weak solution?
• RQ2 Can multiple weak solutions be proven to exist when $ f(u) $ satisfies specific symmetry or growth conditions?
• RQ3 How does the nonlocal interaction term $ \int_{\Omega} \frac{|u|^{2_{\mu}^{\ast}}}{|x-y|^\mu} dy $ affect the solution multiplicity in bounded domains?
• RQ4 What role does the parameter $ \lambda > 0 $ play in the existence and number of solutions?
• RQ5 How do the variational structure and critical exponent interact to influence compactness and existence?

Key findings

• The paper proves the existence of at least one nontrivial weak solution for the Choquard equation under general assumptions on $ f(u) $, including subcritical and asymptotically linear growth.
• When $ f(u) $ satisfies additional symmetry or oscillatory conditions, the equation admits multiple weak solutions, demonstrating solution multiplicity.
• The critical exponent $ 2_{\mu}^{\ast} = \frac{2N - \mu}{N - 2} $ arises naturally from the Hardy-Littlewood-Sobolev inequality and governs the nonlocal interaction strength.
• The variational framework successfully handles the lack of compactness due to the critical exponent and nonlocal term via careful analysis of the Palais-Smale condition.
• The parameter $ \lambda > 0 $ influences the number and nature of solutions, with larger values potentially enabling multiple solutions under suitable $ f(u) $.