[Paper Review] Multiplicity results for the fractional laplacian in expanded domains

Giovany M. Figueiredo, Marcos T. O. Pimenta+1 more|arXiv (Cornell University)|Nov 30, 2015

Nonlinear Partial Differential Equations19 references3 citations

TL;DR

This paper establishes multiplicity results for nontrivial weak solutions to the fractional Laplacian equation $(-\Delta)^\alpha u + u = h(u)$ in expanding domains $\Omega_\lambda = \lambda\Omega$, using variational methods, Ljusternick-Schnirelmann category, and Morse theory. It proves that for large $\lambda$, the number of solutions is bounded below by topological invariants of the domain: at least $\operatorname{cat}(\Omega_\lambda)$ solutions, and $\operatorname{cat}(\Omega_\lambda)+1$ if the domain is not contractible, with improved bounds via Morse theory using the Poincaré polynomial.

ABSTRACT

In this paper we establish the multiplicity of nontrivial weak solutions for the problem $(-Δ)^α u +u= h(u)$ in $Ω_λ$,\ $u=0$ on $\partialΩ_λ$, where $Ω_λ=λΩ$, $Ω$ is a smooth and bounded domain in $\mathbb{R}^N, N>2α$, $λ$ is a positive parameter, $α\in (0,1)$, $(-Δ)^α$ is the fractional Laplacian and the nonlinear term $h(u)$ has a subcritical growth. We use minimax methods, the Ljusternick-Schnirelmann and Morse theories to get multiplicity result depending on the topology of $Ω$.

Motivation & Objective

• To establish the existence and multiplicity of nontrivial weak solutions for the fractional Laplacian equation in expanding domains $\Omega_\lambda = \lambda\Omega$.
• To link the number of solutions to topological invariants of the domain $\Omega_\lambda$, particularly its Ljusternick-Schnirelmann category and Poincaré polynomial.
• To extend previous results on classical Laplacian equations to the fractional setting using variational methods and critical point theory.
• To analyze how the number of solutions grows with the parameter $\lambda$ as the domain expands.
• To provide sharp lower bounds on the number of solutions based on the topology of $\Omega_\lambda$, including non-contractible domains.

Proposed method

• Use of the fractional Laplacian $(-\Delta)^\alpha$ for $\alpha \in (0,1)$ in a bounded, smooth domain $\Omega \subset \mathbb{R}^N$, $N > 2\alpha$.
• Application of variational methods to define an energy functional whose critical points correspond to weak solutions of the equation.
• Employment of the Ljusternick-Schnirelmann category to estimate the number of critical points based on the topology of $\Omega_\lambda$.
• Use of Morse theory with the Poincaré polynomial $\mathcal{P}_1(\Omega_\lambda)$ to obtain improved multiplicity bounds.
• Reduction of the problem to a constrained functional on a manifold $\mathcal{M}_\lambda$ via the Nehari method, ensuring boundedness and regularity of critical points.
• Use of deformation arguments and homology isomorphisms to relate the topology of $\Omega_\lambda$ to the topology of sublevel sets of the energy functional.

Experimental results

Research questions

• RQ1 How does the number of nontrivial weak solutions to the fractional Laplacian equation grow as the domain $\Omega_\lambda$ expands with $\lambda \to \infty$?
• RQ2 Can topological invariants of the domain $\Omega_\lambda$ such as the Ljusternick-Schnirelmann category predict the number of solutions?
• RQ3 What is the role of the nonlinearity $h(u)$ with subcritical growth and the Ambrosetti-Rabinowitz condition in ensuring multiplicity?
• RQ4 Can Morse theory provide a sharper lower bound on the number of solutions than Ljusternick-Schnirelmann theory?
• RQ5 How do the assumptions on $h(u)$, including $h(s)/s$ being increasing and $\theta H(s) \leq s h(s)$, affect the multiplicity of solutions?

Key findings

• For $\lambda \geq \lambda^*$, the problem has at least $\operatorname{cat}(\Omega_\lambda)$ weak solutions, where $\operatorname{cat}$ denotes the Ljusternick-Schnirelmann category.
• If $\Omega_\lambda$ is not contractible, i.e., $\operatorname{cat}(\Omega_\lambda) > 1$, then there are at least $\operatorname{cat}(\Omega_\lambda) + 1$ weak solutions for $\lambda \geq \lambda^*$.
• Under stronger differentiability assumptions on $h$, Morse theory yields at least $2\mathcal{P}_1(\Omega_\lambda) - 1$ solutions for $\lambda > \lambda^*$, where $\mathcal{P}_1$ is the Poincaré polynomial evaluated at $t=1$.
• The number of solutions is strictly determined by the topology of the domain $\Omega_\lambda$, with no dependence on the specific shape beyond homotopy type.
• The results are robust under small perturbations of the domain and rely on the existence of regular regular values and discrete critical sets in the functional framework.
• The multiplicity bounds are sharp in the sense that they are derived from topological invariants and are not improved by further assumptions on $h$ beyond the stated conditions.