In the present paper, we are concerned with the following fractional p-Laplacian Choquard logarithmic equation(−Δ)psu+V(x)|u|p−2u+(ln|⋅|⁎|u|p)|u|p−2u=(∫RNF(y,u)|y|β|x−y|μdy)f(x,u)|x|βinRN, where ...N=sp≥2, s∈(0,1), 0<μ<N, β≥0, 2β+μ≤N and (−Δ)ps denotes the fractional p-Laplace operator, the potential V∈C(RN,0,∞)), and f:RN×R→R is continuous. Under mild conditions and combining variational and topological methods, we obtain the existence of axially symmetric solutions both in the exponential subcritical case and in the exponential critical case. We point out that we take advantage of some refined analysis techniques to get over the difficulty carried by the competition of the Choquard logarithmic term and the Stein-Weiss nonlinearity. Moreover, in the exponential critical case, we extend the nonlinearities to more general cases compared with the existing results.
The World Health Organization declared the coronavirus disease 2019 a pandemic on March 11th, pointing to the over 118,000 cases in over 110 countries and territories around the world at that time. ...At the time of writing this manuscript, the number of confirmed cases has been surging rapidly past the half-million mark, emphasizing the sustained risk of further global spread. Governments around the world are imposing various containment measures while the healthcare system is bracing itself for tsunamis of infected individuals that will seek treatment. It is therefore important to know what to expect in terms of the growth of the number of cases, and to understand what is needed to arrest the very worrying trends. To that effect, we here show forecasts obtained with a simple iteration method that needs only the daily values of confirmed cases as input. The method takes into account expected recoveries and deaths, and it determines maximally allowed daily growth rates that lead away from exponential increase toward stable and declining numbers. Forecasts show that daily growth rates should be kept at least below 5% if we wish to see plateaus any time soon—unfortunately far from reality in most countries to date. We provide an executable as well as the source code for a straightforward application of the method on data from other countries.
This paper is dedicated to studying the existence of normalized solutions for a class of Chern-Simons-Schrödinger system, where the nonlinearity possesses critical exponential growth of ...Trudinger-Moser type. Under some weak assumptions, we obtain several new existence results by employing more delicate estimates and analytical technical. Our results improve and complement the works of Yao et al. (2023) 25 and Yuan et al. (2022) 27.
We investigate the evolution of network effects, specifically focusing on the emerging paradigm of third-generation networks known as group-forming networks (GFNs). In contrast to traditional views ...where network value adheres to Metcalfe’s quadratic growth or Sarnoff’s linear growth, GFNs, such as social networks and community-centric platforms like Apple, Microsoft, and Google, exhibit a distinct exponential growth pattern in network value as postulated by Reed’s hypothesis.
Unlike conventional network science, economic perspectives on network effects do not necessitate precise knowledge of the network topology. Instead, they often align with mean-field models akin to those found in statistical physics. Recent empirical studies challenge earlier assumptions, revealing that many GFNs have undergone exponential or notably superquadratic growth, deviating significantly from established models based solely on user and link counts.
This paper introduces a simplified model aimed at capturing the growth dynamics of marketable groups within a network comprising n users. The model explores various hypothetical mechanisms governing the expansion of these groups, and through rigorous analysis and simulation, we uncover compelling evidence that the observed growth in GFNs surpasses traditional quadratic models. Our findings challenge existing notions, suggesting a new framework for understanding the intricacies of network growth dynamics, and providing solid analytical and conceptual foundations for the recent explosive processes of economic and financial value growth of networks that exploit the formation of groups.
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•Network Effects: Focus on sub-networks and groups, beyond user size and P2P.•Network Value: Considers revenues from groups of dynamic sizes.•Ad Revenue Growth: Finds explosive growth in groups ad revenues beyond standard economic models.•Superlinear Dynamics: explosive growth in group ads drives platform success and de-facto monopolies.
This paper is concerned with the following planar Schrödinger-Poisson system{−Δu+V(x)u+ϕu=f(x,u),x∈R2,Δϕ=u2,x∈R2, where V∈C(R2,0,∞)) is axially symmetric and f∈C(R2×R,R) is of subcritical or critical ...exponential growth in the sense of Trudinger-Moser. We obtain the existence of a nontrivial solution or a ground state solution of Nehari-type and infinitely many solutions to the above system under weak assumptions on V and f. Our theorems extend the results of Cingolani and Weth Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016) 169-197 and of Du and Weth Nonlinearity, 30 (2017) 3492-3515 and Chen and Tang J. Differential Equations, 268 (2020) 945-976, where f(x,u) has polynomial growth on u. In particular, some new tricks and approaches are introduced to overcome the double difficulties resulting from the appearance of both the convolution ϕ2,u(x) with sign-changing and unbounded logarithmic integral kernel and the critical growth nonlinearity f(x,u).
In this paper, we consider the existence of solutions for nonlinear elliptic equations of the form(0.1)−Δu+V(|x|)u=Q(|x|)f(u)+λg(|x|,u),x∈R2, where the nonlinear term f(s) has critical exponential ...growth which behaves like eαs2, g(r,s) is a concave term on s, the radial potentials V,Q:R+→R are unbounded, singular at the origin or decaying to zero at infinity and λ>0 is a parameter. Based on the known Trudinger-Moser inequality in H0,rad1(B1), we establish a new version of Trudinger-Moser inequality in the working space of the associated with the energy functional related to the above problem. By combining the variational methods, Trudinger-Moser inequality and some new approaches to estimate precisely the minimax level of the energy functional, we prove the existence of a nontrivial solution for the above problem under some weak assumptions. Our results show that the presence of the concave term (i.e. λ>0) is very helpful to the existence of nontrivial solutions for Eq. (0.1) in one sense.
Selection among autocatalytic species fundamentally depends on their growth law: exponential species, whose number of copies grows exponentially, are mutually exclusive, while sub-exponential ones, ...whose number of copies grows polynomially, can coexist. Here we consider competitions between autocatalytic species with different growth laws and make the simple yet counterintuitive observation that sub-exponential species can exclude exponential ones while the reverse is, in principle, impossible. This observation has implications for scenarios pertaining to the emergence of natural selection.
What is scaling? Bohan, Sarah; Tippmann, Esther; Levie, Jonathan ...
Journal of business venturing,
January 2024, 2024-01-00, Letnik:
39, Številka:
1
Journal Article
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As ‘scaling’ has gained significant attention from different stakeholders, multiple definitions have emerged, endangering the legitimacy of the area as a distinct field of inquiry. Using a ...mathematics perspective, we define scaling in the business context as a time-limited process of exponential growth. We then identify drivers of scaling and show that scaling for competitive advantage requires increasing returns to scale in input-output relationships (superlinear scaling). This is followed by the application of graph theory, supported with findings from a Delphi study, to demonstrate why scaling requires internal transformation. Finally, we discuss our definition's uniqueness, how it can be operationalized, and opportunities for future research.
•Scaling in the business context is a time-limited phase of exponential growth.•Scaling for competitive advantage requires scaling of input-output relationships.•Digitalization has turbo-charged the drivers of scaling.•Scaling needs proactive, well-paced internal transformation and sustained innovation.
In the present paper, we study the following planar Choquard equation:{−Δu+V(x)u=(Iα⁎F(u))f(u),x∈R2,u∈H1(R2), where V(x) is an 1-periodic function, Iα:R2→R is the Riesz potential and f(t) behaves ...like ±eβt2 as t→±∞. A direct approach is developed in this paper to deal with the problems with both critical exponential growth and strongly indefinite features when 0 lies in a gap of the spectrum of the operator −△+V. In particular, we find nontrivial solutions for the above equation with critical exponential growth, and establish the existence of ground states and geometrically distinct solutions for the equation when the nonlinearity has subcritical exponential growth. Our results complement and generalize the known ones in the literature concerning the positive potential V to the general sign-changing case, such as, the results of de Figueiredo-Miyagaki-Ruf (1995) 16, of Alves-Cassani-Tarsi-Yang (2016) 4, of Ackermann (2004) 1, and of Alves-Germano (2018) 5.
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