The paper considers the following nonhomogeneous SchroedingeraMaxwell system: (SM) { - I u + u + I> I (x) u = | u | p - 1 u + g (x) , x a R 3 , - I I = u 2 , x a R 3 , where I> > 0 , p a (1 , 5) , ...and 0 0 such that problem (SM) has at least two solutions for all p a (1 , 5) provided that a g a L 2 0 is small. Moreover, C p = (p - 1) 2 p (p + 1) S p + 1 2 p 1 / (p - 1) , where S is the Sobolev constant.
In this paper, we consider the existence and multiplicity of solutions of Kirchhoff type problems with critical nonlinearity in R super(3):- epsilon super(2)(a+bint sub(R3) |(grad)u| super(2)dx) ...Delta u+V(x)u=K(x)|u| super(4)u+h(x,u)R3:- epsilon 2(a+bint R3|(grad)u|2dx) Delta u+V (x)u=K(x)|u|4u+h(x,u), (t,x)isinRR super(3)(t,x)isinRR3. Under suitable assumptions, we prove that this has at least one solution and for any misinNmisinN, it has at least mm pairs of solutions.
In this paper, we consider the existence and multiplicity of solutions of Kirchhoff type problems with critical nonlinearity in R super(3) : -epsilon super(2) (a + b integraloperator sub(R) ...super(3)Hami lton operatoru| super(2)dx) Deltau + V(x)u = K(x)|u| super(4)u + h(x, u), (t, x) isin R x R super(3). Under suitable assumptions, we prove that this has at least one solution and for any m isin N, it has at least m pairs of solutions.
We show the existence of two nontrivial nonnegative solutions and infinitely many solutions for degeneratep(x)-Laplace equations involving concave-convex type nonlinearities with two parameters. By ...investigating the order of concave and convex terms and using a variational method, we determine the existence according to the range of each parameter. Some Caffarelli-Kohn-Nirenberg type problems with variable exponents are also discussed.
2010Mathematics Subject Classification: 35J20, 35J60, 35J70, 47J10, 46E35.
Key words and phrases:p(x)-Laplacian,Weighted variable exponent Lebesgue-Sobolev spaces, Concave-convex nonlinearities, Nonnegative solutions, Multiplicity.
Abstract
Retaining an imprint of their thermal history is a hallmark of glassy materials. Although its microscopic origin is still in debate, this memory effect is the potential to be utilized in ...engineering applications as a way to rejuvenate the glasses. For a better understanding of it, we investigated how the memory effect is affected by non-exponentiality and non-linearity, which are two basic features of glass dynamics. A mathematical model with a linear superposition of relaxation functions at a series of experienced temperatures was employed to reproduce the memory effect. The results demonstrate that non-exponentiality has a leading role in determining memory behaviors while non-linearity influences it weakly. An enhanced memory effect found in a recent multistep temperature training experiment is understood with the decreasing non-exponentiality caused by the increasing dynamical heterogeneities of the system. This work provides a guide to regulating the memory effect in practical applications.
In this paper, we consider the following nonlinear Schrödinger equations with mixed nonlinearities:{−Δu=λu+μ|u|q−2u+|u|2⁎−2uin RN,u∈H1(RN),∫RNu2=a2, where N≥3, μ>0, λ∈R and 2<q<2⁎=2NN−2. We prove in ...this paper(1)Existence of solutions of mountain-pass type for N≥3 and 2<q<2+4N;(2)Existence and nonexistence of ground states for 2+4N≤q<2⁎ with μ>0 large;(3)Precisely asymptotic behaviors of ground states and mountain-pass solutions as μ→0 and μ goes to its upper bound. Our studies answer some open questions proposed by Soave in 48.
Passive acoustic foams have limitations in efficiently reducing low-frequency noise below 800 Hz due to the associated wavelengths for reasonable material thickness. To address this challenge, ...passive or semi-active resonators are commonly employed solutions. Linear resonators are very efficient within a narrow frequency bandwidth. However, nonlinear oscillators can broaden this bandwidth, but activation of their nonlinear responses typically requires high excitation amplitudes, beyond human hearing tolerance amplitudes. Furthermore, the specific type of nonlinearity in such devices is often predetermined by the inherent properties of the resonator. In this study, we employ a novel digital control algorithm, allowing to activate the nonlinear response of the electroacoustic resonator at low excitation amplitudes. This algorithm, which relies on real-time integration, facilitates the creation of nonlinear resonators featuring polynomial or diverse non-polynomial nonlinearities within the range of low amplitudes. The nonlinear control is carried out on a loudspeaker equipped with a microphone. Our research highlights the potential to create nonlinear resonators with different, versatile and programmable behaviors. Unprecedented non-polynomial nonlinear behaviors are experimentally exhibited, we consider cubic, piece-wise linear, and logarithmic nonlinearities. These behaviors are implemented and compared to a semi-analytic model to control an acoustic mode of a tube under conditions of low excitation amplitudes and frequencies.
•We exploit advantages of nonlinearities in vibro-acoustical control.•The exploited nonlinearities are polynomial and non-polynomial.•We digitally program nonlinearities of an electroacoustic resonator.•The nonlinear behaviors are activated at low excitation amplitudes.•The programmed resonator is used for control of an acoustic mode.
Optical nonlinearity impairments have been a major obstacle for high-speed, long-haul and large-capacity optical transmission. In this paper, we propose a novel convolutional neural network ...(CNN)-based perturbative nonlinearity compensation approach in which we reconstruct a feature map with two channels that rely on first-order perturbation theory and build a classifier and a regressor as a nonlinear equalizer. We experimentally demonstrate the CNN equalizer in 375 km 120-Gbit/s dual-polarization 64-quadrature-amplitude modulation (64-QAM) coherent optical communication systems. We studied the influence of the dropout value and nonlinear activation function on the convergence of the CNN equalizer. We measured the bit-error-ratio (BER) performance with different launched optical powers. When the channel size is 11, the optimum BER for the CNN classifier is 0.0012 with 1 dBm, and for the CNN regressor, it is 0.0020 with 0 dBm; the BER can be lower than the 7<inline-formula><tex-math notation="LaTeX">\%</tex-math></inline-formula> hard decision-forward threshold of 0.0038 from −3 dBm to 3 dBm. When the channel size is 15, the BERs at −4 dBm, 4 dBm and 5 dBm can be lower than 0.0020. The network complexity is also analyzed in this paper. Compared with perturbative nonlinearity compensation using a fully connected neural network (2392-64-64), we can verify that the time complexity is reduced by about 25<inline-formula><tex-math notation="LaTeX">\%</tex-math></inline-formula>, while the space complexity is reduced by about 50<inline-formula><tex-math notation="LaTeX">\%</tex-math></inline-formula>.
We show the existence of solution in L super(p) spaces for a generalized form of the classical Von Karman equations where the coefficients of nonlinear terms are variable. We use Campanato's near ...operators theory.
