[Paper Review] Existence and symmetric result for Liouville-Weyl fractional nonlinear Schrödinger equation

César E. Torres Ledesma|arXiv (Cornell University)|Dec 4, 2014

Advanced Mathematical Physics Problems22 references3 citations

TL;DR

This paper establishes the existence of positive solutions for a one-dimensional fractional nonlinear Schrödinger equation with mixed Liouville-Weyl fractional derivatives using variational methods and symmetric decreasing rearrangement techniques. The key contribution is proving that ground state solutions are radially symmetric and positive under weaker assumptions on the potential than previously required, extending existence results beyond coercive potentials.

ABSTRACT

We study the existence of positive solution for the one dimensional Schrödinger equation with mixed Lioville-Weyl fractional derivatives \begin{eqnarray*}\label{Eq00} _{t}D_{\infty}^α({_{-\infty}}D_{t}^αu(t)) + V(t) u(t) = & f(u(t)),\;\;t\in \mathbb{R}\\ u\in H^α(\mathbb{R}). onumber \end{eqnarray*} Furthermore, we analyse radial symmetry property of these solutions. The proof is carried out by using variational methods jointly with comparison and rearrangement argument.

Motivation & Objective

• To establish the existence of positive solutions for a one-dimensional Schrödinger equation involving mixed left and right fractional derivatives.
• To analyze the radial symmetry properties of these solutions using rearrangement techniques.
• To weaken the coercivity assumption on the potential $ V(t) $, which is typically required in prior works on fractional boundary value problems.
• To apply critical point theory in a fractional Sobolev space framework for equations with nonlocal operators.
• To extend variational methods to fractional equations with mixed derivatives by constructing a suitable energy functional and verifying compactness conditions.

Proposed method

• Formulates the problem as a variational framework in the fractional Sobolev space $ H^{ au}(ℝ) $ with $ \alpha \in (1/2,1) $.
• Defines a weak solution via a variational formulation involving the inner product of fractional derivatives and the potential term.
• Applies the mountain pass theorem by verifying the geometric conditions for the energy functional $ I(u) $, ensuring a critical level $ c > 0 $.
• Uses symmetric decreasing rearrangement $ u^* $ of a function $ u $ to compare energy levels and prove symmetry of minimizers.
• Employs the fractional Pólya-Szegö inequality to show that $ I(u^*) \leq I(u) $, preserving or reducing energy under rearrangement.
• Combines continuity of the rearrangement map on $ H^\alpha(\mathbb{R}) $ with the concentration-compactness principle to extract a symmetric critical point.

Experimental results

Research questions

• RQ1 Under what conditions does the fractional Schrödinger equation with mixed Liouville-Weyl derivatives admit a positive solution in $ H^\alpha(\mathbb{R}) $?
• RQ2 Can the existence of a ground state solution be established without assuming coercivity of the potential $ V(t) $?
• RQ3 Is the ground state solution radially symmetric and decreasing about the origin under suitable assumptions on $ f $ and $ V $?
• RQ4 How does the symmetric decreasing rearrangement affect the energy functional in the context of fractional Schrödinger equations?
• RQ5 What variational framework allows for the application of critical point theory to equations with nonlocal mixed derivatives?

Key findings

• The energy functional $ I(u) $ satisfies the mountain pass geometry, ensuring the existence of a critical level $ c > 0 $, which corresponds to a nontrivial solution.
• Under assumptions $ (f_0) $--$ (f_3) $, $ (V_1) $, and $ (V_3) $, a sequence of functions $ u_n $ converges strongly in $ H^\alpha(\mathbb{R}) $ to a solution $ u $ with $ I(u) = c $ and $ I'(u) = 0 $.
• The symmetric decreasing rearrangement $ u^* $ of any solution satisfies $ I(u^*) \leq I(u) $, with equality only if $ u $ is already symmetric.
• The ground state solution $ u $ is radially symmetric and decreasing, i.e., $ u(x) = u^*(x) $, and $ u^* $ is a minimizer of $ I $ among all functions with the same $ L^2 $-norm.
• The proof relies on the continuity of the rearrangement map on $ H^\alpha(\mathbb{R}) $, as established by Theorem 5.1, ensuring convergence of rearranged paths.
• The result holds under weaker assumptions on $ V(t) $ than the coercivity condition used in previous works, such as in [22], thus broadening the applicability of variational methods to fractional Schrödinger equations.