Sublinear Term Research Articles

Existence of positive solutions for a class of second-order ordinary differential systems

In this paper, we present some new concepts of superlinearity and sublinearity for nonlinear terms of second-order ordinary differential systems, and then consider the existence of positive solutions for systems with such superlinear and sublinear nonlinear terms, especially proving the existence of positive solutions for systems in which one nonlinear term is superlinear and the other is sublinear.

Channel Coherence in the Low-SNR Regime

Channel capacity in the limit of vanishing signal-to-noise ratio (SNR) per degree of freedom is known to be linear in SNR for fading and nonfading channels, regardless of channel state information at the receiver (CSIR). It has recently been shown that the significant engineering difference between the coherent and the noncoherent fading channels, including the requirement of peaky signaling and the resulting spectral efficiency, is determined by how the capacity limit is approached as SNR tends to zero, or in other words, the sublinear term in the capacity expression. In this paper, we show that this sublinear term is determined by the channel coherence level, which we define to quantify the relation between the SNR and the channel coherence time. This allows us to trace a continuum between the case with perfect CSIR and the case with no CSIR at all. Using this approach, we also evaluate the performance of suboptimal training schemes

Finite difference approximate solutions for a mixed sub-superlinear equation

In Ben Mabrouk and Ben Mohamed (2006) [A. Ben Mabrouk, M.L. Ben Mohamed, On some critical and slightly super-critical sub-superlinear equations. Far East J. Appl. Math. 23(1) (2006) 73–90, Special Volume of PDEs; A. Ben Mabrouk, M.L. Ben Mohamed, Nodal solutions for some nonlinear elliptic equations, Appl. Math. Comp., in press, doi:10.1016/j.amc.2006.08.003], it has proved some theoretical results dealing with some boundary value problem Δ u + f( u) = 0 in B and u = 0 on ∂ B, where B is a domain in R d with smooth boundary. In the present paper, we perform a difference scheme method to approximate the solution of a nonlinear evolutionary problem associated to the elliptic problem studied in Ben Mabrouk and Ben Mohamed (2006). We give at the end some numerical implementations based on a mixed sublinear superlinear term of the form f( u) = ∣ u∣ p−1 u + λ∣ u∣ q−1 u.

Ground state solutions for the singular Lane–Emden–Fowler equation with sublinear convection term

We are concerned with singular elliptic equations of the form − Δ u = p ( x ) ( g ( u ) + f ( u ) + | ∇ u | a ) in R N ( N ⩾ 3 ), where p is a positive weight and 0 < a < 1 . Under the hypothesis that f is a nondecreasing function with sublinear growth and g is decreasing and unbounded around the origin, we establish the existence of a ground state solution vanishing at infinity. Our arguments rely essentially on the maximum principle.

Positive solutions of critical semilinear problems involving a sublinear term at the origin

By using variational methods we prove that a class of critical semilinear problems with a sublinear term at the origin possesses at least two positive solutions under conditions which allow the primitive of the sublinear term to assume negative values and do not require a convexity assumption. These problems are treated on bounded domains in R N , N > 2.

On a singular sublinear polyharmonic problem

This paper deals with a class of singular nonlinear polyharmonic equations on the unit ballBinℝn (n≥2)where the combined effects of a singular and a sublinear term allow us by using the Schauder fixed point theorem to establish an existence result for the following problem:(−Δ)mu=φ(⋅,u)+ψ(⋅,u)inB(in the sense of distributions),u&gt;0,lim⁡|x|→1u(x)/(1−|x|)m−1=0. Our approach is based on estimates for the polyharmonic Green function onBwith zero Dirichlet boundary conditions.

The Nehari manifold for a semilinear elliptic equation involving a sublinear term

The Nehari manifold for the equation $$ -\Delta u(x) = \lambda u(x) + b(x) \vert u(x)\vert^{\gamma - 2} u(x) $$ for $ x \in \Omega $ together with Dirichlet boundary conditions is investigated in the case where $ 1 < \gamma < 2$ . Exploiting the relationship between the Nehari manifold and fibrering maps (i.e., maps of the form $t \longrightarrow J(tu)$ where J is the Euler functional associated with the equation), we discuss how the Nehari manifold changes as $\lambda$ changes and show how this is linked to results on bifurcation from infinity which are associated with the problem.

Solutions of Hammerstein Integral Equations via a Variational Principle

We study solutions of the nonlinear Hammer- stein integral equation with changing-sign kernels by using a variational principle of Ricceri and critical points theory tech- niques. Combining the effects of a sublinear and superlinear nonlinear terms we establish new existence and multiplicity results for the equation. As an application we consider a semi- linear Dirichlet problem for polyharmonic elliptic operators.

Structure of ground state solutions of singular semilinear elliptic equations

This paper deals with a class of singular semilinear elliptic equations in R N where the combined effects of a singular and a sublinear term allow us to establish some existence, symmetry and uniqueness results for positive “ground state” solutions.

Small Periodic Solutions Generated by Sublinear Terms

The paper is concerned with Hopf bifurcations in systems of autonomous ordinary differential equations with a parameter. The principal distinction between usual theorems on Hopf bifurcations and our results is that here the linearized equation is degenerate and independent of the parameter. We present sufficient conditions for a parameter value to be a bifurcation point and analyze properties of small cycles arising in the vicinity of the equilibrium. Sublinear nonlinearities play the main role in the results obtained.

Cycle Stability for Hopf Bifurcation Generated by Sublinear Terms

The paper is concerned with the study of small stable cycles of autonomous quasilinear systems depending on a parameter. Sufficient conditions are presented for the existence of such cycles for control theory equations with scalar nonlinearities. The principal distinction of the case considered from usual results on Hopf bifurcations is that the linear part of the problem is degenerate for all the parameter values (not only at a bifurcation point). Small sublinear nonlinearities play the main role in our results. The proofs are based on the theory of monotone operators.

A sign-changing solution for elliptic equation with sublinear term at origin

It is proved that the semilinear elliptic problem with zero boundary value −Δu=λu−|u| q−1 u has a changing sign solution, as q∈(0,1) and λ>λ2, where λ2 is the second eigenvalue of the operator −Δ in the space H 0 1 (Ω).

Existence of critical points in a general setting

The paper proves the existence of critical points for a general locally Lipschitz functional usually arising in nonlinear elliptic problems. It extends and unifies various results in the critical point theory. The applications treat new situations involving discontinuous elliptic equations containing both sublinear and superlinear terms, integro-differential equations and nonlinear elliptic systems.