[Paper Review] Positive solutions to some asymptotically linear fractional Schrodinger equations

Jinguo Zhang, Xiaochun Liu|arXiv (Cornell University)|Nov 9, 2014

Nonlinear Partial Differential Equations6 references3 citations

TL;DR

This paper establishes the existence of positive solutions for a class of fractional Schrödinger equations with asymptotically linear nonlinearity in $\mathbb{R}^N$, $N>2s$, using variational methods and a new representation of Palais-Smale sequences. The key contribution is proving that under suitable conditions on the potential $V(x)$ and nonlinearity $f(u)$, a critical point exists in the energy functional, yielding at least one positive solution despite the lack of the Ambrosetti-Rabinowitz condition.

ABSTRACT

This paper is devoted to prove the existence and nonexistence of positive solutions for a class of fractional Schrodinger equation in RN of the We apply a new methods to obtain the existence of positive solutions when f(u) is asymptotically linear with respect to u at infinity.

Motivation & Objective

• To establish the existence of positive solutions for fractional Schrödinger equations with asymptotically linear nonlinearity in $\mathbb{R}^N$, where $N>2s$ and $s\in(0,1)$.
• To address the challenge of non-compact embeddings in $H^s(\mathbb{R}^N)$ and the lack of symmetry in the problem.
• To overcome the failure of the Ambrosetti-Rabinowitz condition, which is commonly used in variational methods but does not hold for asymptotically linear nonlinearities.
• To develop a new representation theorem for Palais-Smale sequences, identifying the only compactness obstacle as solutions to the limiting problem at infinity.
• To prove that the infimum of the energy functional is not a critical value, and under an additional condition, that a critical point exists, yielding a positive solution.

Proposed method

• The study employs variational methods on the space $H^s(\mathbb{R}^N)$, using the energy functional $\mathcal{I}(u) = \frac{1}{2}\int |(-\Delta)^{s/2}u|^2 dx + \frac{1}{2}\int V(x)|u|^2 dx - \int F(u) dx$.
• A new representation theorem for Palais-Smale sequences is derived, showing that the only compactness obstruction arises from solutions to the limiting equation at infinity: $(-\Delta)^s u + V_\infty u = f(u)$.
• The Pohozaev identity is applied to analyze the behavior of solutions and to derive necessary conditions for nonexistence of solutions.
• A linking theorem is used in conjunction with a deformation lemma to prove the existence of a critical point in the energy functional, under the condition $V_5$.
• The proof relies on constructing a suitable deformation retract $Q = \Gamma(\overline{B}_{\bar{\rho}}(0))$ and a set $S = \{u \in \mathcal{P} : \beta(u) = 0\}$, showing that they link in the sense of the linking theorem.
• The analysis includes careful estimates of the potential term $V(x) - V_\infty$ and the use of scaling arguments to control the behavior of rescaled functions.

Experimental results

Research questions

• RQ1 Under what conditions does a fractional Schrödinger equation with asymptotically linear nonlinearity admit a positive solution in $\mathbb{R}^N$?
• RQ2 How can one overcome the lack of compactness in $H^s(\mathbb{R}^N)$ embeddings when the nonlinearity is asymptotically linear and the Ambrosetti-Rabinowitz condition fails?
• RQ3 What is the role of the limiting problem at infinity in obstructing compactness of Palais-Smale sequences?
• RQ4 Can the infimum of the energy functional be a critical value when the nonlinearity is asymptotically linear and the (AR)-condition is not satisfied?
• RQ5 What geometric and analytic conditions on the potential $V(x)$ ensure the existence of a positive solution?

Key findings

• The infimum $p = \inf\{\mathcal{I}(u) : u \in \mathcal{P}\}$ is not a critical value of the energy functional $\mathcal{I}$, and thus is not achieved.
• Under the additional condition $(V_5)$, which bounds the ratio $\frac{(N-2s)\|w\|^2_{H^s}}{A\|w\|^2_{L^2}} \leq 2^{2s/N}$ with $A = \max_x \langle \nabla V(x), x \rangle$, the problem admits at least one positive solution in $H^s(\mathbb{R}^N)$.
• The critical level $d$ of the energy functional satisfies $m_\infty < d < 2m_\infty$, where $m_\infty$ is the infimum of the energy on the limiting problem at infinity.
• The Palais-Smale condition $(PS)_d$ holds in the interval $(m_\infty, 2m_\infty)$, ensuring the existence of a critical point via the linking theorem.
• The solution obtained is positive due to the maximum principle, under the given assumptions on $f(u)$, including $f(t)/t \to 1$ as $t \to \infty$ and $f(t)/t \to 0$ as $t \to 0^+$.
• The analysis shows that the only compactness obstruction in the Palais-Smale sequence is the presence of nontrivial solutions to the limiting equation at infinity, which is rigorously controlled through scaling and norm estimates.