Ground states for diffusion system with periodic and asymptotically periodic nonlinearity

Zhang, Jian; Tang, Xianhua; Wen, Zhang

doi:10.1016/j.camwa.2015.12.031

Cited by 9 publications

(3 citation statements)

References 47 publications

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“…Because this identity provides no information for non-autonomous systems, the approach in [27] is no longer applicable for problem (1.1) with non-autonomous nonlinearity. For other related problems, we refer to [31][32][33][34][35][36][37] and so on.…”

Section: Introductionmentioning

confidence: 99%

Ground state solutions for asymptotically periodic fractional Schrödinger-Poisson problems with asymptotically cubic or super-cubic nonlinearities

Chen

Peng

Tang

2017

Math. Meth. Appl. Sci.

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In this paper, we consider the following fractional Schrödinger–Poisson problem: (−△)su+V(x)u+K(x)ϕ(x)u=f(x,u),1emx∈double-struckR3,(−△)tϕ=K(x)u2,1emx∈double-struckR3, where s,t∈(0,1],4s+2t>3,V(x),K(x), and f(x,u) are periodic or asymptotically periodic in x. We use the non‐Nehari manifold approach to establish the existence of the Nehari‐type ground state solutions in two cases: the periodic one and the asymptotically periodic case, by introducing weaker conditions lim|τ|→∞∫0τf(x,ξ)normaldξ|τ|σ=∞ uniformly in x∈double-struckR3 with σ:=max{3,4−2t} and f(x,τ)τ3−f(x,kτ)(kτ)3sign(1−k)+θ0V(x)|1−k2|(kτ)2⩾0,∀x∈R3,k>0,τ≠0 with constant θ0∈(0,1), instead of lim|τ|→∞∫0τf(x,ξ)normaldξ|τ|4=∞ uniformly in x∈double-struckR3 and the usual Nehari‐type monotonic condition on f(x,τ)/|τ|3. Our results unify both asymptotically cubic or super‐cubic nonlinearities, which are new even for s=t=1. Copyright © 2017 John Wiley & Sons, Ltd.

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Section: Introductionmentioning

confidence: 99%

Ground state solutions for asymptotically periodic fractional Schrödinger-Poisson problems with asymptotically cubic or super-cubic nonlinearities

Chen

Peng

Tang

2017

Math. Meth. Appl. Sci.

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“…This type of problem arises in many different applications, such as continuum mechanics, phase transition phenomena, population dynamics, and game theory, as they are the typical outcome of stochastically stabilization of Lévy processes; see [1][2][3][4][5][6][7][8][9] and the references therein. The literature on nonlocal operators and their applications is very interesting and quite large; we refer the interested reader to [4,[10][11][12][13][14][15][16][17][18][19][20][21] and the references therein. For the basic properties of fractional Sobolev spaces, we refer the interested reader to [22,23].…”

Section: Introductionmentioning

confidence: 99%

Nontrivial Solution for the Fractional p-Laplacian Equations via Perturbation Methods

Luo¹,

Li²,

Tang³

2017

Advances in Mathematical Physics

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We study the existence of nontrivial solution of the following equation without compactness:where , ≥ 2, ∈ (0, 1), (−Δ) is the fractional -Laplacian, and the subcritical -superlinear term ∈ (R × R) is 1-periodic in for = 1, 2, . . . , . Our main difficulty is that the weak limit of (PS) sequence is not always the weak solution of fractional -Laplacian type equation. To overcome this difficulty, by adding coercive potential term and using mountain pass theorem, we get the weak solution of perturbation equations. And we prove that → as → 0. Finally, by using vanishing lemma and periodic condition, we get that is a nontrivial solution of fractional -Laplacian equation.

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“…By applying the generalized linking theorems in [36], the existence of 'the least energy solutions' (i.e. a minimizer of the corresponding energy within the set of nontrivial solutions) or multiple solutions were obtained for problem (1) with periodic potential and nonlinearity, see [16,18,22,24,26,[28][29][30][31][38][39][40]. For the non-periodic case, we refer the readers to [20,21,25,32,41] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Ground state solutions of Nehari‐Pankov type for a superlinear elliptic system on

Tang

Zhang

2016

Math Methods in App Sciences

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This paper is concerned with the following Schrödinger elliptic system −normalΔu+V(x)u=g(x,v),1emx∈double-struckRN,−normalΔv+V(x)v=f(x,u),1emx∈double-struckRN,u(x)→01emand1emv(x)→0,1em1em1em1emas1em|x|→∞, where the potential V is periodic and 0 lies in a gap of the spectrum of −Δ+V, f(x,t) and g(x,t) are superlinear in t at infinity. By using non‐Nehari manifold method developed recently by Tang, we demonstrate the existence of the ground state solutions of Nehari‐Pankov type for the above problem with periodic and asymptotically periodic nonlinearity. Copyright © 2016 John Wiley & Sons, Ltd.

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Ground states for diffusion system with periodic and asymptotically periodic nonlinearity

Cited by 9 publications

References 47 publications

Ground state solutions for asymptotically periodic fractional Schrödinger-Poisson problems with asymptotically cubic or super-cubic nonlinearities

Ground state solutions for asymptotically periodic fractional Schrödinger-Poisson problems with asymptotically cubic or super-cubic nonlinearities

Nontrivial Solution for the Fractional p-Laplacian Equations via Perturbation Methods

Ground state solutions of Nehari‐Pankov type for a superlinear elliptic system on

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