[Paper Review] Solutions for fractional operator problem via local Pohozaev identities

Yuxia Guo, Ting Liu+1 more|arXiv (Cornell University)|Apr 17, 2019

Nonlinear Partial Differential Equations28 references4 citations

TL;DR

This paper establishes the existence of infinitely many solutions for a fractional Schrödinger equation with critical exponent using a finite-dimensional reduction method combined with local Pohozaev identities. Under a weaker symmetry condition on the potential $ V(|y'|, y'') $, it proves that multi-bump solutions emerge via precise asymptotic analysis and perturbation techniques near non-degenerate critical points of $ r^{2s}V(r, y'') $.

ABSTRACT

We consider the following fractional Schrödinger equation involving critical exponent: \begin{equation*} \left\{\begin{array}{ll} (-Δ)^s u+V(|y'|,y'')u=u^{2^*_s-1} \ \hbox{ in } \ \mathbb{R}^N, \\ u>0, \ y \in \mathbb{R}^N, \end{array} ight. \end{equation*} where $s\in(\frac{1}{2}, 1)$, $(y',y'')\in \mathbb{R}^2 imes \mathbb{R}^{N-2}$, $V(|y'|,y'')$ is a bounded nonnegative function with a weaker symmetry condition. We prove the existence of infinitely many solutions for the above problem by a finite dimensional reduction method combining various Pohazaev identies.

Motivation & Objective

• To establish the existence of infinitely many positive solutions for a fractional Schrödinger equation with critical nonlinearity in $ \mathbb{R}^N $.
• To address the challenge of nonlocality in the fractional Laplacian $ (-\Delta)^s $ with $ s \in (\frac{1}{2}, 1) $, particularly in the context of critical exponent problems.
• To extend previous results on multi-bump solutions by relaxing the symmetry assumptions on the potential $ V(y) $, allowing for weaker conditions than radial or full rotational symmetry.
• To develop a refined analytical framework combining finite-dimensional reduction with local Pohozaev identities to control the asymptotic behavior of solutions near concentration points.

Proposed method

• Utilizes a finite-dimensional reduction method to reduce the infinite-dimensional variational problem to a finite-dimensional one near approximate solutions constructed from cutoff bubbles $ Z_{x_j,\lambda} $.
• Employs local Pohozaev identities to derive precise asymptotic expansions of the gradient of the energy functional, enabling control over the location and scale parameters of the approximate solutions.
• Constructs a family of approximate solutions $ Z_{\bar{r},\bar{y}^{\prime\prime},\lambda} $ by modifying the standard ground state $ U_{x,\lambda} $ using a cutoff function $ \zeta $, ensuring compact support and decay in the energy space.
• Applies weighted norms and estimates in the space $ H^s(\mathbb{R}^N) $ to control the error terms arising from the nonlinearity and the potential $ V $, particularly in the region where $ V $ is non-zero.
• Implements a Lyapunov-Schmidt type decomposition to solve the projected equation, reducing the problem to a finite-dimensional system in the parameters $ \bar{r}, \bar{y}^{\prime\prime}, \lambda $.
• Uses the degree theory and non-degeneracy of the critical point of $ r^{2s}V(r, y^{\prime\prime}) $ to ensure the existence of a solution to the reduced system, thereby yielding a true solution to the original equation.

Experimental results

Research questions

• RQ1 Can infinitely many solutions be constructed for the fractional Schrödinger equation with critical exponent when the potential $ V $ satisfies only a weaker symmetry condition?
• RQ2 How can local Pohozaev identities be adapted to handle the nonlocal nature of the fractional Laplacian in a multi-bump solution setting?
• RQ3 What is the role of the function $ r^{2s}V(r, y^{\prime\prime}) $ in determining the location and stability of multi-bump solutions?
• RQ4 Under what conditions does the finite-dimensional reduction method remain valid for the fractional Schrödinger equation with critical growth?
• RQ5 How do the decay properties of the bubble solutions $ U_{x,\lambda} $ and their cutoff versions $ Z_{x_j,\lambda} $ affect the convergence and existence of solutions in the energy space?

Key findings

• The paper proves the existence of infinitely many positive solutions to the fractional Schrödinger equation with critical exponent $ 2^*_s = \frac{2N}{N-2s} $ in $ \mathbb{R}^N $, under the condition that $ r^{2s}V(r, y^{\prime\prime}) $ has a non-degenerate critical point with non-zero degree.
• The solutions are constructed as multi-bump solutions, each concentrating near the vertices of a regular $ k $-gon in the $ y' $-plane, with the $ j $-th bump centered at $ x_j = (\bar{r} \cos \theta_j, \bar{r} \sin \theta_j, \bar{y}^{\prime\prime}) $, where $ \theta_j = \frac{2(j-1)\pi}{k} $.
• The asymptotic error in the gradient of the energy functional is shown to be $ O\left(\frac{k}{\lambda^{2s+1+\sigma}}\right) $, which is sufficiently small to allow for a Lyapunov-Schmidt reduction to succeed.
• The method relies on the fact that the perturbation terms from the potential and nonlinearity are controlled via weighted $ L^2 $-type estimates and pointwise decay of the cutoff bubbles.
• The critical point of $ r^{2s}V(r, y^{\prime\prime}) $ at $ (r_0, y_0^{\prime\prime}) $ with non-zero degree ensures the existence of a solution to the reduced finite-dimensional problem.
• The analysis confirms that the local Pohozaev identities are essential in capturing the correct balance between the nonlocal operator, the potential, and the nonlinearity, especially in the regime where $ N \leq 6s $.