Ground state solutions of Nehari–Pohozaev type for Kirchhoff-type problems with general potentials

Math Methods in App Sciences

2019

In this paper, we study the existence and multiplicity of standing waves with prescribed L2‐norm Schrödinger‐Poisson equations with general nonlinearities in double-struckR3: i∂tψ+Δψ−κ(|x|−1*|φ|2)ψ+f(ψ)=0, where κ>0 and f is superlinear and satisfies the monotonicity condition. To this end, we look for critical points of the following functional Eκ(u)=12∫R3|∇u|2+κ4∫R3(|x|−1*u2)u2−∫R3F(u) constrained on the L2‐spheres Sfalse(cfalse)={}u∈H1false(double-struckR3false):false|false|ufalse|false|22=c,1emc>0, where Ffalse(sfalse):=∫0sffalse(tfalse)dt. We consider the case where Eκ is unbounded from below on Sfalse(cfalse) and establish the existence of critical points of Eκ on Sfalse(cfalse) for c>0 sufficiently small and under some mild assumptions on f. In addition, we show that there are infinitely many radial critical points false{unκfalse} of Eκ on Sfalse(cfalse) when f is odd and present a convergence property of unκ as κ↘0. Our results generalize some recent results in the literature.

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

“…

x

{double-struckR}^{3}

. Motivated by Tang and Chen and Xie and Chen, denote

w_{n} = v_{n} - v

, then we have

E false(v_{n} false) = E false(w_{n} false) + E false(v false) + o false(1 false) and J false(v_{n} false) = J false(w_{n} false) + J false(v false) + o false(1 false)

and

normalΨ false(w_{n} false) = m false(c false) - normalΨ false(v false) + o false(1 false), J false(w_{n} false) = - J false(v false) + o false(1 false),

where

normalΨ false(u false) = E false(u false) - \frac{1}{2} J false(u false)

. We may assume that

w_{n} \neq 0

for all

n \in double-struckN

.…”

Section: Proof Of Theorems 1 Andmentioning

confidence: 99%

Existence and multiplicity of normalized solutions for a class of Schrödinger‐Poisson equations with general nonlinearities

Xie

Math Methods in App Sciences

2019

“…Later, Ruiz [11] proved the existence of a positive radial solution for 3 < p ≤ 4 by introducing the Nehari-Pohozaev manifold and establishing a key inequality. For more results on the Schrödinger-Maxwell system and related systems, we refer the readers to [12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27]. In [28], Azzollini and Pomponio obtained the existence of a ground state solution for the subcritical cases 3 < p < 5 and the critical case g(…”

Section: Q(y)f(u(y))mentioning

confidence: 99%

Ground state solutions for asymptotically periodic Schrödinger–Poisson systems involving Hartree-type nonlinearities

Wen

2018

Bound Value Probl

Self Cite

“…In real life, proportional delay plays a huge role in many areas such as quality of web, current collection [24] and so on. Since the applications of CNNs have an important relation with the global exponential convergence behaviors [25][26][27][28][29][30][31][32][33][34][35][36][37]. Therefore we think that it is meaningful to analyze the global exponential convergence of neural networks with proportional delays and leakage delays.…”

Section: Introductionmentioning

confidence: 99%

Dynamics of FCNNs with proportional delays and leakage delays

Guo

et al. 2018

Adv Differ Equ

In this article, a kind of fuzzy cellular neural networks (FCNNs) with proportional delays and leakage delays are involved. Utilizing the differential inequality strategies, a sequence of sufficient criteria ensuring the global exponential convergence of involved model are presented. Computer simulations are performed to verify the analytic findings. The analytic findings of this article are innovative and complete several existing works.