Fractional Hamiltonian systems with critical exponential growth

Nonlinear Differential Equations and Applications NoDEA

In this paper, we study the following nonlocal nonautonomous Hamiltonian system on whole R

We study a class of nonlinear nonautonomous nonlocal equations with subcritical and critical exponential nonlinearity. The involved potential can vanish at infinity.

Electronic Journal of Differential Equations, 2022

In this article, we show the existence of positive solution to the nonlocal system (−∆) s u + a(x)u = 1 2 * s Hu(u, v) in R N , (−∆) s v + b(x)v = 1 2 * s Hv(u, v) in R N , u, v > 0 in R N , u, v ∈ D s,2 (R N). We also prove a global compactness result for the associated energy functional similar to that due to Struwe in [26]. The basic tools are some information from a limit system with a(x) = b(x) = 0, a variant of the Lion's principle of concentration and compactness for fractional systems, and Brouwer degree theory.

In this chapter, we will discuss three-dimensional discrete dynamical systems with long-range properties. The continuum limits for equations of discrete dy-namical systems are used to obtain equations for distributed systems that exibit a power-law non-locality. Some applications for diierent types of dynamical systems are discussed. Fractional generalizations of the nonlinear equations such as sine-Gordon, Burgers, Korteweg-deVries, Kadomtsev-Petviashvili, Boussinesq, and Navier-Stokes equations for 3D lattice with long-range properties and nonlocal continuum are considered.

Mathematics

This work aims to analyze a new system of two fractional Hamiltonian equations. We propose an effective method for transforming the established model into a system of two distinct equations. Two functionals that are connected to the converted system of fractional Hamiltonian systems are introduced together with a new space, and it is demonstrated that these functionals are bounded below on this space. The hypotheses presented here differ from those provided in the literature.

Science China Mathematics

In this paper we study the nonhomogeneous semilinear fractional Schrödinger equation with critical growth    (−∆) s u + u = u 2 * s −1 + λ(f (x, u) + h(x)), x ∈ R N , u ∈ H s (R N), u(x) > 0, x ∈ R N , where s ∈ (0, 1), N > 4s, and λ > 0 is a parameter, 2 * s = 2N N −2s is the fractional critical Sobolev exponent, f and h are some given functions. We show that there exists 0 < λ * < +∞ such that the problem has exactly two positive solutions if λ ∈ (0, λ *), no positive solutions for λ > λ * , a unique solution (λ * , u λ *) if λ = λ * , which shows that (λ * , u λ *) is a turning point in H s (R N) for the problem. Our proofs are based on the variational methods and the principle of concentration-compactness.

Advanced Nonlinear Studies, 2016

We consider a fractional Schrödinger–Poisson system with a general nonlinearity in the subcritical and critical case. The Ambrosetti–Rabinowitz condition is not required. By using a perturbation approach, we prove the existence of positive solutions. Moreover, we study the asymptotics of solutions for a vanishing parameter.

In this work we want to prove the existence of solution for a class of fractional Hamiltonian systems given by {eqnarray*}_{t}D_{\infty}^{\alpha}(_{-\infty}D_{t}^{\alpha}u(t)) + L(t)u(t) = & \nabla W(t,u(t)) u\in H^{\alpha}(\mathbb{R}, \mathbb{R}^{N}) {eqnarray*}

In this paper, we investigate the local existence and the finite-time blow-up of solutions of semilinear parabolic system with nonlocal in time nonlinearity. In addition, we also give the blow-up rate and necessary conditions for local and global existence.

Fractal and fractional, 2022

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