Multiplicity of Solutions to a Concave–Convex Problem

2015 ◽  
Vol 15 (1) ◽  
Author(s):  
José C. Sabina de Lis ◽  
Sergio Segura de León
Keyword(s):  
Smooth Domain ◽  
Multiplicity Of Solutions ◽  
Convex Problem ◽  
Degenerate Regime

AbstractThis paper studies two related problems. A first one,where Δwhere Ω is a ball and 1 < q < p < r. Features described above are shown to be also exhibited by (Q) and more importantly, it is proved that minimal solutions to (Q) develop flat patterns in the degenerate regime p > 2. Finally, it should be stressed that some of the properties satisfied by (P) and (Q) hold true when Ω is a general smooth domain.

scholarly journals High perturbations of a new Kirchhoff problem involving the p-Laplace operator

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Zhongyi Zhang ◽  
Yueqiang Song
Keyword(s):  
Laplace Operator ◽  
Smooth Domain ◽  
Multiplicity Of Solutions ◽  
Bounded Smooth Domain ◽  
Existence And Multiplicity ◽  
Bounded Smooth ◽  
Nonlocal Terms ◽  
Kirchhoff Problem

AbstractIn the present work we are concerned with the existence and multiplicity of solutions for the following new Kirchhoff problem involving the p-Laplace operator: $$ \textstyle\begin{cases} - (a-b\int _{\Omega } \vert \nabla u \vert ^{p}\,dx ) \Delta _{p}u = \lambda \vert u \vert ^{q-2}u + g(x, u), & x \in \Omega , \\ u = 0, & x \in \partial \Omega , \end{cases} $$ { − ( a − b ∫ Ω | ∇ u | p d x ) Δ p u = λ | u | q − 2 u + g ( x , u ) , x ∈ Ω , u = 0 , x ∈ ∂ Ω , where $a, b > 0$ a , b > 0 , $\Delta _{p} u := \operatorname{div}(|\nabla u|^{p-2}\nabla u)$ Δ p u : = div ( | ∇ u | p − 2 ∇ u ) is the p-Laplace operator, $1 < p < N$ 1 < p < N , $p < q < p^{\ast }:=(Np)/(N-p)$ p < q < p ∗ : = ( N p ) / ( N − p ) , $\Omega \subset \mathbb{R}^{N}$ Ω ⊂ R N ($N \geq 3$ N ≥ 3 ) is a bounded smooth domain. Under suitable conditions on g, we show the existence and multiplicity of solutions in the case of high perturbations (λ large enough). The novelty of our work is the appearance of new nonlocal terms which present interesting difficulties.

scholarly journals Existence and Multiplicity of Solutions for Semipositone Problems Involvingp-Laplacian

2012 ◽  
Vol 2012 ◽  
pp. 1-10
Author(s):  
Guowei Dai ◽  
Chunfeng Yang
Keyword(s):  
Positive Solutions ◽  
Smooth Domain ◽  
Multiplicity Of Solutions ◽  
Bounded Smooth Domain ◽  
Existence And Multiplicity ◽  
Bounded Smooth ◽  
Multiplicity Of Positive Solutions

We prove existence and multiplicity of positive solutions for semipositone problems involvingp-Laplacian in a bounded smooth domain ofℝNunder the cases of sublinear and superlinear nonlinearities term.

Sum-Rate Optimization Constrained by Consumed Power for Multi- Antenna Non-Regenerative Relay Network

2020 ◽  
Vol 10 (2) ◽  
pp. 143-152 ◽  
Author(s):  
Maryam Alibeigi ◽  
Shahriar S. Moghaddam
Keyword(s):  
Relay Network ◽  
Minimization Algorithm ◽  
Amplify And Forward ◽  
Relay Nodes ◽  
Convex Problem ◽  
Semidefinite Programming Problem ◽  
Total Leakage ◽  
Sum Rate ◽  
Rate Optimization ◽  
Novel Algorithm

Background & Objective: This paper considers a multi-pair wireless network, which communicates peer-to-peer using some multi-antenna amplify-and-forward relays. Maximizing the throughput supposing that the total relay nodes’ power consumption is constrained, is the main objective of this investigation. We prove that finding the beamforming matrix is not a convex problem. Methods: Therefore, by using a semidefinite relaxation technique we find a semidefinite programming problem. Moreover, we propose a novel algorithm for maximizing the total signal to the total leakage ratio. Numerical analyses show the effectiveness of the proposed algorithm which offers higher throughput compared to the existing total leakage minimization algorithm, with much less complexity. Results and Conclusion: Furthermore, the effect of different parameters such as, the number of relays, the number of antennas in each relay, the number of transmitter/receiver pairs and uplink and downlink channel gains are investigated.

scholarly journals Physical layer security transmission scheme based on artificial noise in cooperative SWIPT NOMA system

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Yong Jin ◽  
Zhentao Hu ◽  
Dongdong Xie ◽  
Guodong Wu ◽  
Lin Zhou
Keyword(s):  
Interference Cancellation ◽  
Physical Layer Security ◽  
High Energy ◽  
Physical Layer ◽  
Base Station ◽  
Artificial Noise ◽  
Transmitted Power ◽  
Transmission Scheme ◽  
Convex Problem ◽  
Security Problem

AbstractAiming at high energy consumption and information security problem in the simultaneous wireless information and power transfer (SWIPT) multi-user wiretap network, we propose a user-aided cooperative non-orthogonal multiple access (NOMA) physical layer security transmission scheme to minimize base station (BS) transmitted power in this paper. In this scheme, the user near from BS is adopted as a friendly relay to improve performance of user far from BS. An energy harvesting (EH) technology-based SWIPT is employed at the near user to collect energy which can be used at cooperative stage. Since eavesdropper in the downlink of NOMA system may use successive interference cancellation (SIC) technology to obtain the secrecy information of receiver, to tackle this problem, artificial noise (AN) is used at the BS to enhance security performance of secrecy information. Moreover, semidefinite relaxation (SDR) method and successive convex approximation (SCA) technique are combined to solve the above non-convex problem. Simulation results show that in comparison with other methods, our method can effectively reduce the transmitted power of the BS on the constraints of a certain level of the secrecy rates of two users.

scholarly journals Existence and multiplicity of solutions for a fractional p-Laplacian equation with perturbation

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Zhen Zhi ◽  
Lijun Yan ◽  
Zuodong Yang
Keyword(s):  
Variational Methods ◽  
Bounded Domain ◽  
Analytical Techniques ◽  
Nontrivial Solutions ◽  
Multiplicity Results ◽  
Multiplicity Of Solutions ◽  
Laplacian Equation ◽  
Existence And Multiplicity

AbstractIn this paper, we consider the existence of nontrivial solutions for a fractional p-Laplacian equation in a bounded domain. Under different assumptions of nonlinearities, we give existence and multiplicity results respectively. Our approach is based on variational methods and some analytical techniques.

scholarly journals Joint optimization of computing ratio and access points’ density for mixed mobile edge/cloud computing

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Tianqi Jing ◽  
Shiwen He ◽  
Fei Yu ◽  
Yongming Huang ◽  
Luxi Yang ◽  
...  
Keyword(s):  
Cloud Computing ◽  
Coverage Probability ◽  
Optimization Problem ◽  
Mobile Cloud Computing ◽  
Low Complexity ◽  
Mixed Integer ◽  
Joint Optimization ◽  
Mobile Cloud ◽  
Convex Problem

AbstractCooperation between the mobile edge computing (MEC) and the mobile cloud computing (MCC) in offloading computing could improve quality of service (QoS) of user equipments (UEs) with computation-intensive tasks. In this paper, in order to minimize the expect charge, we focus on the problem of how to offload the computation-intensive task from the resource-scarce UE to access point’s (AP) and the cloud, and the density allocation of APs’ at mobile edge. We consider three offloading computing modes and focus on the coverage probability of each mode and corresponding ergodic rates. The resulting optimization problem is a mixed-integer and non-convex problem in the objective function and constraints. We propose a low-complexity suboptimal algorithm called Iteration of Convex Optimization and Nonlinear Programming (ICONP) to solve it. Numerical results verify the better performance of our proposed algorithm. Optimal computing ratios and APs’ density allocation contribute to the charge saving.

Sign in / Sign up

Export Citation Format

Share Document