[Paper Review] Solutions for a nonlocal elliptic equation involving critical growth and Hardy potential

Chunhua Wang, Jing Yang+1 more|arXiv (Cornell University)|Sep 24, 2015

Nonlinear Partial Differential Equations31 references3 citations

TL;DR

This paper establishes the existence of infinitely many solutions for a nonlocal elliptic equation involving critical Sobolev growth and Hardy potential in a bounded domain containing the origin. Using an approximating argument and variational methods in the fractional Sobolev space $ H^s_0(Ω) $, the authors prove that under the condition $ N > 6s $, the problem admits infinitely many solutions when the Hardy coefficient $ \mu $ satisfies a specific spectral threshold condition and $ a > 0 $.

ABSTRACT

In this paper, by an approximating argument, we obtain infinitely many solutions for the following Hardy-Sobolev fractional equation with critical growth \begin{equation*}\label{0.1} \left\{% \begin{array}{ll} (-Δ)^{s} u-\ds\frac{μu}{|x|^{2s}}=|u|^{2^*_s-2}u+au, & \hbox{$ ext{in}~ Ω$},\vspace{0.1cm} u=0,\,\, &\hbox{$ ext{on}~\partial Ω$}, \\ \end{array}% ight. \end{equation*} provided $N>6s$, $μ\geq0$, $0< s<1$, $2^*_s=\frac{2N}{N-2s}$, $a>0$ is a constant and $Ω$ is an open bounded domain in $\R^N$ which contains the origin.

Motivation & Objective

• To establish the existence of infinitely many solutions for a nonlocal elliptic equation with critical Sobolev growth and Hardy potential in a bounded domain containing the origin.
• To analyze the interplay between the fractional Laplacian, Hardy potential $ \mu |x|^{-2s} $, and critical nonlinearity $ |u|^{2^*_s - 2}u $ in the context of variational methods.
• To extend previous results on the fractional Laplacian with Hardy potential by incorporating critical growth and proving multiplicity of solutions.
• To investigate the asymptotic behavior of approximating solutions in $ H^s_0(\Omega) $, particularly their decomposition into localized profiles.
• To verify that the spectral threshold condition on $ \mu $ ensures the validity of the associated Hardy inequality and the coercivity of the energy functional.

Proposed method

• Formulates the problem as a variational framework using the energy functional $ I(u) = \frac{1}{2}\int_\Omega \left( |(-\Delta)^{s/2}u|^2 - \mu \frac{u^2}{|x|^{2s}} - a u^2 \right) dx - \frac{1}{2^*_s}\int_\Omega |u|^{2^*_s} dx $ on $ H^s_0(\Omega) $.
• Employs an approximating argument by introducing a vanishing parameter $ \epsilon_n \to 0 $, solving a regularized version of the equation and analyzing the limit.
• Applies the concentration-compactness principle and profile decomposition to analyze the asymptotic structure of bounded sequences in $ H^s_0(\Omega) $, decomposing solutions into a limit profile and localized bubbles.
• Uses the spectral definition of the fractional Laplacian $ (-\Delta)^s $ via eigenfunction expansion of the Laplacian on $ \Omega $, ensuring well-definedness in $ H^s_0(\Omega) $.
• Relies on the sharp Hardy inequality involving the optimal constant $ \bar{\mu} = 2^{2s} \Gamma^2(\frac{N+2s}{4}) / \Gamma^2(\frac{N-2s}{4}) $, which controls the singularity at the origin.
• Applies the method of moving planes and Pohozaev-type identities in the extension problem to derive energy identities and control boundary terms in the blow-up analysis.

Experimental results

Research questions

• RQ1 Under what conditions on $ \mu $, $ s $, and $ N $ does the fractional elliptic equation with Hardy potential and critical nonlinearity admit infinitely many solutions?
• RQ2 How does the presence of the Hardy potential $ \mu |x|^{-2s} $ affect the variational structure and compactness of Palais-Smale sequences?
• RQ3 What is the asymptotic profile of approximating solutions as the regularization parameter $ \epsilon_n \to 0 $, and how do they decompose into localized bubbles?
• RQ4 Can the energy functional associated with the equation support infinitely many critical points when $ N > 6s $, even with a nontrivial Hardy potential?
• RQ5 What role does the spectral threshold condition $ \frac{2^*_s \sqrt{\bar{\mu}}}{\sqrt{\bar{\mu}} - \sqrt{\bar{\mu} - \mu}} > \frac{2N}{N - 6s} $ play in ensuring the existence of multiple solutions?

Key findings

• The equation admits infinitely many weak solutions in $ H^s_0(\Omega) $ for $ N > 6s $, $ \mu \geq 0 $, $ a > 0 $, and $ \mu $ satisfying the spectral threshold condition.
• The approximating solutions $ u_n $ decompose as $ u_n = u_0 + \sum_{j=1}^m \rho_{0,\Lambda_{n,j}}(U_j) + \sum_{j=m+1}^h \rho_{x_{n,j},\Lambda_{n,j}}(U_j) + \omega_n $, where $ \omega_n \to 0 $ in $ H^s(\Omega) $ and $ u_0 $ is a solution of the original problem.
• The localized profiles $ U_j $ are solutions to limiting equations: $ (-\Delta)^s U_j - \mu \frac{U_j}{|x|^{2s}} = b_j |U_j|^{2^*_s - 2} U_j $ in $ D^s(\mathbb{R}^N) $ for $ j=1,\dots,m $, and $ (-\Delta)^s U_j = b_j |U_j|^{2^*_s - 2} U_j $ for $ j=m+1,\dots,h $, with $ b_j \in (0,1] $.
• The concentration points $ x_{n,j} $ for $ j > m $ satisfy $ \Lambda_{n,j} d(x_{n,j}, \partial\Omega) \to \infty $ and $ \Lambda_{n,j} |x_{n,j}| \to \infty $, indicating they escape to the boundary or infinity.
• The mutual separation condition $ \frac{\Lambda_{n,j}}{\Lambda_{n,i}} + \frac{\Lambda_{n,i}}{\Lambda_{n,j}} + \Lambda_{n,j} \Lambda_{n,i} |x_{n,i} - x_{n,j}|^2 \to \infty $ holds for $ i \neq j $, ensuring orthogonality of bubbles in the limit.
• The energy functional is bounded from below and satisfies the Palais-Smale condition along the approximating sequence, which is essential for the existence of infinitely many solutions.