1,720,975 research outputs found
Normalized ground states for the NLS equation with combined nonlinearities
Full text linkWe study existence and properties of ground states for the nonlinear Schrödinger equation with combined power nonlinearities −Δu=λu+μ|u|q−2u+|u|p−2uin RN, N≥1, having prescribed mass ∫RN|u|2=a2. Under different assumptions on q0 and μ∈R we prove several existence and stability/instability results. In particular, we consider cases when [Formula presented] i.e. the two nonlinearities have different character with respect to the L2-critical exponent. These cases present substantial differences with respect to purely subcritical or supercritical situations, which were already studied in the literature. We also give new criteria for global existence and finite time blow-up in the associated dispersive equation
Normalized ground states for the NLS equation with combined nonlinearities: The Sobolev critical case
Full text linkWe study existence and properties of ground states for the nonlinear Schrödinger equation with combined power nonlinearities −Δu=λu+μ|u|q−2u+|u|2javax.xml.bind.JAXBElement@4d419c48−2uin RN, N≥3, having prescribed mass ∫RN|u|2=a2, in the Sobolev critical case. For a L2-subcritical, L2-critical, of L2-supercritical perturbation μ|u|q−2u we prove several existence/non-existence and stability/instability results. This study can be considered as a counterpart of the Brezis-Nirenberg problem in the context of normalized solutions, and seems to be the first contribution regarding existence of normalized ground states for the Sobolev critical NLSE in the whole space RN
The nodal set of solutions to some elliptic problems: Singular nonlinearities
Full text linkThis paper deals with solutions to the equation-Delta u = lambda(+ )(u(+))(q-1) - lambda(-)(u(-))(q-1 )in B-1where lambda(+), lambda(-) > 0, q is an element of (0, 1), B-1 = B-1(0) is the unit ball in R-N, N >= 2, and u(+) := maxu, 0, u(-) := max-u, 0 are the positive and the negative part of u, respectively. We extend to this class of singular equations the results recently obtained in [25] for sublinear and discontinuous equations, 1 <= q < 2, namely: (a) the finiteness of the vanishing order at every point and the complete characterization of the order spectrum; (b) a weak non-degeneracy property; (c) regularity of the nodal set of any solution: the nodal set is a locally finite collection of regular codimension one manifolds up to a residual singular set having Hausdorff dimension at most N - 2 (locally finite when N = 2). As an intermediate step, we establish the regularity of a class of not necessarily minimal solutions.The proofs are based on a priori bounds, monotonicity formulae for a 2-parameter family of Weiss-type functionals, blow-up arguments, and the classification of homogeneous solutions. (C) 2019 Elsevier Masson SAS. All rights reserved
A fountain of positive bubbles on a Coron's problem for a competitive weakly coupled gradient system
No full textWe consider the following critical elliptic system: {−Δui=μiui3+βui∑j≠iuj2inΩεui=0 on ∂Ωε,ui>0 in Ωεi=1,...,m, in a domain Ωε⊂R4 with a small shrinking hole Bε(ξ0). For μi>0, β<0, and ε>0 small, we prove the existence of a non-synchronized solution which looks like a fountain of positive bubbles, i.e. each component ui exhibits a towering blow-up around ξ0 as ε→0. The proof is based on the Ljapunov-Schmidt reduction method, and the velocity of concentration of each layer within a given tower is chosen in such a way that the interaction between bubbles of different components balances the interaction of the first bubble of each component with the boundary of the domain, and in addition is dominant when compared with the interaction of two consecutive bubbles of the same component
Local minimizers in absence of ground states for the critical NLS energy on metric graphs
Full text linkWe consider the mass-critical non-linear Schrödinger equation on non-compact metric graphs. A quite complete description of the structure of the ground states, which correspond to global minimizers of the energy functional under a mass constraint, is provided by Adami, Serra and Tilli in [R. Adami, E. Serra and P. Tilli. Negative energy ground states for the L2-critical NLSE on metric graphs. Comm. Math. Phys. 352 (2017), 387-406.], where it is proved that existence and properties of ground states depend in a crucial way on both the value of the mass, and the topological properties of the underlying graph. In this paper we address cases when ground states do not exist and show that, under suitable assumptions, constrained local minimizers of the energy do exist. This result paves the way to the existence of stable solutions in the time-dependent equation in cases where the ground state energy level is not achieved
On the effectiveness of heat-exchanger bypass control
Full text linkIn heat exchangers with bypassing, a fraction of the flowrate of one fluid (typically the one whose temperature needs to be controlled tightly) bypasses the exchanger and mixes right after the exchanger outlet with the fraction flowing through the exchanger. The advantages of this configuration are long known. Among them, the most significant is that it can improve heat-transfer control because the temperature dynamics is significantly faster than in a standard heat-exchanger configuration. Additionally, it can increase the rangeability of the process wherein the heat exchanger operates. Existing rules of thumb do not provide univocal indications for assigning the design bypass flowrate. In this study, using a simple graphical representation of steady-state heat and mass balances originally proposed for conventional heat-exchanger design, we clarify why and under which design conditions bypass control can be effective. Increased rangeability results from the fact that the heat-exchanger steady-state gain can be assigned by design when a bypass configuration is used, whereas it typically cannot in a conventional heat exchanger. The design bypass flowrate should therefore be assigned so as to make the heat exchanger operate in a region where the steady-state gain is relatively high (and constant)
On unique continuation principles for some elliptic systems
No full textIn this paper we prove unique continuation principles for some systems of elliptic partial differential equations satisfying a suitable superlinearity condition. As an application, we obtain nonexistence of nontrivial (not necessarily positive) radial solutions for the Lane-Emden system posed in a ball, in the critical and supercritical regimes. Some of our results also apply to general fully nonlinear operators, such as Pucci's extremal operators, being new even for scalar equations
Going Beyond Counting First Authors in Author Co-citation Analysis
Full text linkThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
On Coron's problem for weakly coupled elliptic systems
Full text linkWe consider the following critical weakly coupled elliptic system -Îui=Î1⁄4i|ui|2-2ui+jâ iÎ2ij|uj|22|ui|2-42uiinΩÏμui=0onâΩÏμ,i=1,m,in a domain ΩÏμââN, N=3,4, with small shrinking holes as the parameter Ïμâ0. We prove the existence of positive solutions of two different types: either each density concentrates around a different hole, or we have groups of components such that all the components within a single group concentrate around the same point, and different groups concentrate around different points
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