Ground state sign-changing solutions for Kirchhoff type problems in bounded domains

Tang, Xianhua; Cheng, Bitao

doi:10.1016/j.jde.2016.04.032

Cited by 220 publications

(92 citation statements)

References 36 publications

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“…Proof Analogous to the proof of [21], it is easy to show this lemma by combining the quantitative deformation lemma and degree theory.…”

Section: Lemma 28 Assume That (V) and (F1)-(f4) Hold Then Any Sequementioning

confidence: 99%

“…Furthermore, we reduce the usual Nehari-type monotonic condition to a weaker condition (F4). In fact, there are many functions satisfying (F1)-(F4), but not satisfying (F3 ) and (F4 ), and some typical examples are introduced in [21].…”

Section: Remark 13 Since the Term (-)mentioning

confidence: 99%

“…[13][14][15][16][17][18][19][20][21][22]. When a = 1 and b = 0, then (1.1) reduces to the following fractional Schrödinger problem:…”

Section: Introductionmentioning

confidence: 99%

“…Inspired by the previously mentioned work, especially [21,42,43], we seek definite answers to overcome the above three difficulties. Firstly, we use a new trick to show the following set:…”

Section: Introductionmentioning

confidence: 99%

“…Secondly, for the asymptotically periodic case, we adopt a technique introducing in [21] by combining the quantitative deformation lemma and degree theory, to overcome the difficulties caused by the dropping of periodicity of V (x) and f (x, u) in x (see Section 4). Thirdly, because the argument based on Nehari manifold approach (see Szulkin and Weth [44]) becomes invalid under condition (F4) instead of (F4 ), we will use the non-Nehari manifold approach developed in [42,43].…”

Section: Introductionmentioning

confidence: 99%

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Nehari-type ground state solutions for asymptotically periodic fractional Kirchhoff-type problems in RN$\mathbb{R}^{N}$

2018

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In this paper, we studied the following fractional Kirchhoff-type equation:where a, b are positive constants, α ∈ (0, 1), N ∈ (2α, 4α), (-) α is the fractional Laplacian operator, V(x) and f (x, u) are periodic or asymptotically periodic in x. Under some weaker conditions on the nonlinearity, we obtain the existence of ground state solutions for the above problem in periodic case and asymptotically periodic case, respectively. In particular, our results unify both asymptotically cubic and super-cubic nonlinearities, which are new even for α = 1. MSC: 35J20; 35J25; 35J60

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“…Proof Analogous to the proof of [21], it is easy to show this lemma by combining the quantitative deformation lemma and degree theory.…”

Section: Lemma 28 Assume That (V) and (F1)-(f4) Hold Then Any Sequementioning

confidence: 99%

Section: Remark 13 Since the Term (-)mentioning

confidence: 99%

“…[13][14][15][16][17][18][19][20][21][22]. When a = 1 and b = 0, then (1.1) reduces to the following fractional Schrödinger problem:…”

Section: Introductionmentioning

confidence: 99%

“…Inspired by the previously mentioned work, especially [21,42,43], we seek definite answers to overcome the above three difficulties. Firstly, we use a new trick to show the following set:…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Nehari-type ground state solutions for asymptotically periodic fractional Kirchhoff-type problems in RN$\mathbb{R}^{N}$

2018

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Existence of positive solutions for a critical fractional Kirchhoff equation with potential vanishing at infinity

Tang

Yang

2021

Mathematische Nachrichten

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In this paper, we consider the following fractional Kirchhoff equation with critical nonlinearity trueright150.0pt()a+b∫R3||(−Δ)s2u2dxfalse(−normalΔfalse)su+V(x)u=Q(x)f(u)+false|ufalse|2s∗−2u,0.16emx∈R3,where a,b>0, false(−normalΔfalse)s is the fractional Laplace operator with s∈(34,1), V vanishes at infinity and 2s∗=63−2s. Under appropriate assumptions on f, we prove the existence of positive solution by using the variational method.

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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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Ground state sign-changing solutions for Kirchhoff type problems in bounded domains

Cited by 220 publications

References 36 publications

Nehari-type ground state solutions for asymptotically periodic fractional Kirchhoff-type problems in RN$\mathbb{R}^{N}$

Nehari-type ground state solutions for asymptotically periodic fractional Kirchhoff-type problems in RN$\mathbb{R}^{N}$

Existence of positive solutions for a critical fractional Kirchhoff equation with potential vanishing at infinity

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

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