On fractional Schrödinger equation with periodic and asymptotically periodic conditions

Zhang, Wen; Zhang, Jian; Mi, Heilong

doi:10.1016/j.camwa.2017.06.017

Cited by 30 publications

(14 citation statements)

References 37 publications

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“…Motivated by this physical interest, fractional p ‐Laplacian–type problems have been extensively investigated in recent years, and many existence results have been obtained under various hypotheses on the potential and the nonlinearity. We just quote here some recent works . Recently, Souza studied the existence of solutions for the following equation:

{false(- normalΔ false)}_{p}^{s} u + V false(x false) false| u |^{p - 2} u = f false(x, u false) + λ h in R^{N},

using an appropriate Trudinger‐Moser inequality for fractional Sobolev spaces and assuming that f ( x , t ) may be discontinuous and exhibit exponential growth.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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On a class of p‐fractional Laplacian equations with potential depending on parameter

Massar

Talbi²

2019

Math Methods in App Sciences

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We discuss the existence of solutions for a class of nonlinear equations driven by the fractional p‐Laplacian operator of the form (−Δ)psu+λsV(x,λ)|ufalse|p−2u=f(x,u)indouble-struckRN, where 0ps and λ is a positive parameter. By combining variational techniques with a version of the mountain pass theorem without Palais‐Small (PS) condition, we establish the existence of nontrivial solutions for the above equation under certain appropriate assumptions on nonlinearity and weight functions.

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{false(- normalΔ false)}_{p}^{s} u + V false(x false) false| u |^{p - 2} u = f false(x, u false) + λ h in R^{N},

using an appropriate Trudinger‐Moser inequality for fractional Sobolev spaces and assuming that f ( x , t ) may be discontinuous and exhibit exponential growth.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…We just quote here some recent works. [8][9][10][11][12][13][14][15][16][17][18] Recently, Souza 16 studied the existence of solutions for the following equation:…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

On a class of p‐fractional Laplacian equations with potential depending on parameter

Massar

Talbi²

2019

Math Methods in App Sciences

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“…As a motivation we recall that there are many works devoted to the study on the nonlinear Schrödinger equation arising in the non-relativistic quantum mechanics See, for example [1,2,18,21,22,25,26,27,29,38,41] and the references therein. Especially, [1,2,18,26,27,29,38,41] considered the asymptotically periodic case, and obtained the existence results of solutions by means of various variational techniques. Motivated by the above facts, it is quite natural to ask if certain similar results can be forwarded to system (1.1).…”

Section: Remarkmentioning

confidence: 99%

Existence and decay property of ground state solutions for Hamiltonian elliptic system

Zhang¹,

Zhang²

2019

Communications on Pure &Amp; Applied Analysis

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In this paper we study the following nonlinear Hamiltonian elliptic system with gradient term −∆u + b(x) • ∇u + u + V (x)v = f (x, |z|)v, x ∈ R N , −∆v − b(x) • ∇v + v + V (x)u = f (x, |z|)u, x ∈ R N , where z = (u, v) ∈ R 2. Under some suitable conditions on the potential and nonlinearity, we obtain the existence of ground state solutions in periodic case and asymptotically periodic case via variational methods, respectively. Moreover, we also explore some properties of these ground state solutions, such as compactness of set of ground state solutions and exponential decay of ground state solutions.

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“…the fractional Schrödinger equation has become an important model in fractional quantum mechanics. Recently, the fractional Schrödinger equation has attracted a large num-ber of studies for its property, a lot of papers deal with the existence of solutions of it by using critical point theory, see [3][4][5][6][7][8] and the references therein. When s = 1, the fractional Schrödinger equation (1.1) reduces to the following classical nonlinear Schrödinger equation:…”

Section: Introductionmentioning

confidence: 99%

Infinitely many solutions for fractional Schrödinger equation with potential vanishing at infinity

Yun

Zuo

et al. 2019

Bound Value Probl

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The paper investigates the following fractional Schrödinger equation: (-) s u + V(x)u = K(x)f (u), x ∈ R N , where 0 < s < 1, 2s < N, (-) s is the fractional Laplacian operator of order s. V(x), K(x) are nonnegative continuous functions and f (x) is a continuous function satisfying some conditions. The existence of infinitely many solutions for the above equation is presented by using a variant fountain theorem, which improves the related conclusions on this topic. The interesting result of this paper is the potential V(x) vanishing at infinity, i.e., lim |x|→+∞ V(x) = 0.

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On fractional Schrödinger equation with periodic and asymptotically periodic conditions

Cited by 30 publications

References 37 publications

On a class of p‐fractional Laplacian equations with potential depending on parameter

On a class of p‐fractional Laplacian equations with potential depending on parameter

Existence and decay property of ground state solutions for Hamiltonian elliptic system

Infinitely many solutions for fractional Schrödinger equation with potential vanishing at infinity

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