[Paper Review] Multiplicity results for $(p,\, q)$ fractional elliptic equations involving critical nonlinearities

Mousomi Bhakta, Debangana Mukherjee|arXiv (Cornell University)|Jan 30, 2018

Nonlinear Partial Differential Equations21 citations

TL;DR

This paper establishes the existence of infinitely many nontrivial solutions for a class of $(p,q)$ fractional elliptic equations with concave-critical nonlinearities in bounded domains using variational methods. For convex-critical nonlinearities, it proves the existence of at least $\text{cat}_{\Omega}(\Omega)$ nonnegative solutions via a concentration-compactness approach and critical point theory.

ABSTRACT

In this paper we prove the existence of infinitely many nontrivial solutions for the class of $(p,\, q)$ fractional elliptic equations involving concave-critical nonlinearities in bounded domains in $\mathbb{R}^N$. Further, when the nonlinearity is of convex-critical type, we establish the multiplicity of nonnegative solutions using variational methods. In particular, we show the existence of at least $cat_Ω(Ω)$ nonnegative solutions.

Motivation & Objective

• To establish the existence of infinitely many nontrivial solutions for $(p,q)$ fractional elliptic equations with concave-critical nonlinearities in bounded domains.
• To prove the existence of multiple nonnegative solutions when the nonlinearity is of convex-critical type.
• To extend variational methods to the nonlocal $(p,q)$ fractional Laplacian setting in bounded domains.
• To apply concentration-compactness principles and critical point theory to overcome lack of compactness in fractional Sobolev embeddings.
• To connect the multiplicity of solutions to the topological category $\text{cat}_{\Omega}(\Omega)$ of the domain $\Omega$.

Proposed method

• Utilizes the fractional Sobolev space $X_{0,s_1,p}(\Omega)$ and the associated norm involving nonlocal Gagliardo seminorms.
• Applies the concentration-compactness principle to handle lack of compactness in the embedding of $X_{0,s_1,p}(\Omega)$ into $L^{p^*_{s_1}}(\Omega)$.
• Employs variational methods by analyzing the energy functional associated with the equation and proving the (PS) condition below a critical threshold.
• Uses the Krasnoselskii genus and deformation lemma to establish the existence of multiple critical points.
• Applies the mountain pass lemma and deformation arguments to locate critical points of the energy functional.
• Relies on topological tools such as the category $\text{cat}_{\Omega}(\Omega)$ to estimate the number of solutions.

Experimental results

Research questions

• RQ1 Can we prove the existence of infinitely many nontrivial solutions for $(p,q)$ fractional elliptic equations with concave-critical nonlinearities in bounded domains?
• RQ2 What is the precise number of nonnegative solutions when the nonlinearity is of convex-critical type?
• RQ3 How does the topology of the domain $\Omega$ influence the number of solutions to the $(p,q)$ fractional elliptic problem?
• RQ4 Can the concentration-compactness method be adapted to the nonlocal $(p,q)$ fractional setting to recover compactness?
• RQ5 What is the role of the fractional Laplacian operators $(-\Delta)^{s_1}_p$ and $(-\Delta)^{s_2}_q$ in ensuring multiplicity of solutions?

Key findings

• The equation $(P_{\theta,\lambda})$ admits infinitely many nontrivial solutions when the nonlinearity is of concave-critical type.
• For convex-critical nonlinearities, the equation has at least $\text{cat}_{\Omega}(\Omega)$ nonnegative solutions.
• The energy functional satisfies the (PS) condition below the critical level $\frac{s_1}{N}(S_{s_1,p})^{N/(s_1 p)}$, enabling application of critical point theory.
• The existence of multiple solutions is linked to the topological category of the domain $\Omega$, with the number of solutions bounded below by $\text{cat}_{\Omega}(\Omega)$.
• The concentration-compactness method successfully controls the lack of compactness in the fractional Sobolev embedding $X_{0,s_1,p}(\Omega) \hookrightarrow L^{p^*_{s_1}}(\Omega)$.
• The proof relies on constructing a deformation retraction and using odd maps to relate the category of sublevel sets to the number of critical points.