Superlinear Growth Research Articles

On a type of superlinear growth variational problems

In this note, we propose an elementary method to study the existence and uniqueness of solutions to a type of variational problems which arise naturally in the theory of large deviations. This type of problems involves a movable boundary and may not have the coercivity condition in general. Our method is elementarily based on direct analysis over the space of absolutely continuous functions and specific properties of the underlying functional.

English

In this paper, we study the following quasilinear Schrödinger equations: (P) where are given potentials, is a small parameter, g is a even function with and for all and satisfies superlinear growth at infinity. We get the existence results of multiplicity of nontrivial solutions for problem

Partial regularity for symmetric quasiconvex functionals on BD

We establish the first partial regularity results for (strongly) symmetric quasiconvex functionals of linear growth on BD, the space of functions of bounded deformation. By Rindler's foundational work [64], symmetric quasiconvexity is the pivotal notion as regards sequential weak*-lower semicontinuity and hence for the existence of minima of the relaxed functionals on BD. The overarching main difficulty here is the lack of Korn's Inequality in the L1-setting, hereby implying that the BD-case is genuinely different from the study of variational integrals on BV. Unlike for superlinear growth, symmetric quasiconvex functionals, where we establish partial regularity by direct reduction to the full gradient case by Korn-type inequalities, such a reduction does not work in the linear growth case and identifies the latter as the only situation requiring a treatment on its own.

Asymptotically linear and superlinear elliptic equations with gradient terms

In this article we establish the existence of solutions for elliptic problem involving a gradient term. To handle the so-called non-variational problem, we use a variational methods. We assume that the nonlinear term satisfies an asymptotically linear growth condition or a superlinear growth condition. We show the existence of at least one positive solution and one negative solution.
 For more information see https://ejde.math.txstate.edu/Volumes/2020/99/abstr.html

Social cohesion emerging from a community-based physical activity program: A temporal network analysis.

Community-based physical activity programs, such as the Recreovía, are effective in promoting healthy behaviors in Latin America. To understand Recreovías’ challenges and scalability, we characterized its social network longitudinally while studying its participants’ social cohesion and interactions. First, we constructed the Main network of the program’s Facebook profile in 2013 to determine the main stakeholders and communities of participants. Second, we studied the Temporal network growth of the Facebook profiles of three Recreovía locations from 2008 to 2016. We implemented a Time Windows in Networks algorithm to determine observation periods and a scaling model of cities’ growth to measure social cohesion over time. Our results show physical activity instructors as the main stakeholders (20.84% nodes of the network). As emerging cohesion, we found: (1) incremental growth of Facebook users (43–272 nodes), friendships (55–2565 edges), clustering coefficient (0.19–0.21), and density (0.04–0.07); (2) no preferential attachment behavior; and (3) a social cohesion super-linear growth with 1.73 new friendships per joined user. Our results underscore the physical activity instructors’ influence and the emergent cohesion in innovation periods as a co-benefit of the program. This analysis associates the social and healthy behavior dimensions of a program occurring in natural environments under a systemic approach.

Splitting-type variational problems with mixed linear-superlinear growth conditions

Splitting-type variational problems with mixed linear-superlinear growth conditions are considered. In the two-dimensional case the minimization problem is given byJ[w]=∫Ω[f1(∂1w)+f2(∂2w)]dx→min with respect to a suitable class of comparison functions. Here f1 is a convex energy density with linear growth and f2 is a convex energy density with superlinear growth, for instance given by an N-function or just bounded from below by an N-function. One motivation for this kind of problem located between the well-known splitting-type problems of superlinear growth and the splitting-type problems of linear growth (recently considered by M. Bildhauer and M. Fuchs (2020) [8]) is the link to mathematical problems in plasticity (see R. Temam (2018) [29]). Here we prove results in the appropriate way of relaxation, including approximation procedures, duality, and existence and uniqueness of solutions as well as some new higher-integrability results.

Reaction-diffusion systems with initial data of low regularity

Models issued from ecology, chemical reactions and several other application fields lead to semi-linear parabolic equations with super-linear growth. Even if, in general, blow-up can occur, these models share the property that mass control is essential. In many circumstances, it is known that this L1 control is enough to prove the global existence of weak solutions. The theory is based on basic estimates initiated by M. Pierre and collaborators, who have introduced methods to prove L2 a priori estimates for the solution.Here, we establish such a key estimate with initial data in L1 while the usual theory uses L2. This allows us to greatly simplify the proof of some results. We also establish new existence results of semilinearity which are super-quadratic as they occur in complex chemical reactions. Our method can be extended to semi-linear porous medium equations.

On strong convergence of explicit numerical methods for stochastic delay differential equations under non-global Lipschitz conditions

In this paper, we study the convergence of explicit numerical methods in strong sense for stochastic delay differential equations (SDDEs) with super-linear growth coefficients. Under non-globally Lipschitz conditions, a fundamental theorem on convergence has been constructed to elaborate the relationship of convergence rate between the local truncated error and the global error of one-step explicit methods in the sense of pth moments. A class of balanced Euler schemes has been presented and the boundedness of numerical solutions has been proved. By using the fundamental theorem, we prove that the balanced Euler scheme is of 0.5 order convergence in mean-square sense. Numerical examples verify the theoretical predictions.

Ambrosetti\u2013Prodi Type Results for Dirichlet Problems of Fractional Laplacian-Like Operators

We establish Ambrosetti–Prodi type results for viscosity and classical solutions of nonlinear Dirichlet problems for fractional Laplace and comparable operators. In the choice of nonlinearities we consider semi-linear and super-linear growth cases separately. We develop a new technique using a functional integration-based approach, which is more robust in the non-local context than a purely analytic treatment.

Competitive productivity and the challenge of metastasis under rising societal complexity

PurposeThe purpose of this paper is to locate the concept of competitive productivity (CP) within a general theory of societal progress and include new thinking on the challenge of obstacles to be met at certain stages.Design/methodology/approachThe approach is to review the key literature dealing with economic growth and rising societal achievement and to refine out concepts that offer understanding of the dynamics commonly involved, taking illustrative examples from different societies and seeking overall common denominators that appear within the historical processes.FindingsNew understandings of societal progress, using complex adaptive systems theory applied to cities and industrial districts, indicate that two forces are at work to release new positive forms of energy into society. Economies of scale work via the laws of fractal geometry to yield sublinear growth of energy. More intense social interaction works within the core of the society in a different way to yield superlinear growth. These two forms of energy release can feed off each other beneficially in conditions where, as with CP, the forces of competition can work with forces driving efficiency, in conditions where societal order can be supported by appropriate cultural norms.Research limitations/implicationsA wide literature across several disciplines is brought to bear on the very complex question. Some of the theories are new but very well anchored. It is consequently possible to suggest a pattern of multi-determinants able to match the reality and to foster nuanced comprehensive analysis.Practical implicationsImpacts on policy of foreign direct investment and joint venture management.Social implicationsEmphasis on the roles of societal virtues in establishing the cooperativeness needed for CP.Originality/valueFew studies bring together so many disciplinary perspectives into a complete argument.

Inequalities of Green\u2019s functions and positive solutions to nonlocal boundary value problems

In this paper, we discuss the positive solutions of beam equations with the nonlinearities including the slope and bending moment under nonlocal boundary conditions involving Stieltjes integrals. We pose some inequality conditions on nonlinearities and the spectral radius conditions on associated linear operators. These conditions mean that the nonlinearities have superlinear or sublinear growth. The existence of positive solutions is obtained by fixed point index on cones in C^{2}[0,1], and some examples are given for beam equations subject to mixed integral and multi-point boundary conditions with sign-changing coefficients.

Positive periodic solutions to an indefinite Minkowski-curvature equation

We investigate the existence, non-existence, multiplicity of positive periodic solutions, both harmonic (i.e., T-periodic) and subharmonic (i.e., kT-periodic for some integer k≥2) to the equation(u′1−(u′)2)′+λa(t)g(u)=0, where λ>0 is a parameter, a(t) is a T-periodic sign-changing weight function and g:[0,+∞[→[0,+∞[ is a continuous function having superlinear growth at zero. In particular, we prove that for both g(u)=up, with p>1, and g(u)=up/(1+up−q), with 0≤q≤1<p, the equation has no positive T-periodic solutions for λ close to zero and two positive T-periodic solutions (a “small” one and a “large” one) for λ large enough. Moreover, in both cases the “small” T-periodic solution is surrounded by a family of positive subharmonic solutions with arbitrarily large minimal period. The proof of the existence of T-periodic solutions relies on a recent extension of Mawhin's coincidence degree theory for locally compact operators in product of Banach spaces, while subharmonic solutions are found by an application of the Poincaré–Birkhoff fixed point theorem, after a careful asymptotic analysis of the T-periodic solutions for λ→+∞.

Existence and qualitative theory for nonlinear elliptic systems with a nonlinear interface condition used in electrochemistry

We study a nonlinear elliptic system with prescribed inner interface conditions. These models are frequently used in physical system where the ion transfer plays the important role for example in modelling of nano-layer growth or Li-on batteries. The key difficulty of the model consists of the rapid or very slow growth of nonlinearity in the constitutive equation inside the domain or on the interface. While on the interface, one can avoid the difficulty by proving a kind of maximum principle of a solution, inside the domain such regularity for the flux is not available in principle since the constitutive law is discontinuous with respect to the spatial variable. The key result of the paper is the existence theory for these problems, where we require that the leading functional satisfies either the delta-two or the nabla-two condition. This assumption is applicable in case of fast (exponential) growth as well as in the case of very slow (logarithmically superlinear) growth.

Symmetry properties of positive solutions for fully nonlinear elliptic systems

We investigate symmetry properties of positive solutions for fully nonlinear uniformly elliptic systems, such asFi(x,Dui,D2ui)+fi(x,u1,…,un,Dui)=0,1≤i≤n, in a bounded domain Ω in RN with Dirichlet boundary condition u1=…,un=0 on ∂Ω. Here, fi's are nonincreasing with the radius r=|x|, and satisfy a cooperativity assumption. In addition, each fi is the sum of a locally Lipschitz with a nondecreasing function in the variable ui, and may have superlinear gradient growth. We show that symmetry occurs for systems with nondifferentiable fi's by developing a unified treatment of the classical moving planes method in the spirit of Gidas-Ni-Nirenberg. We also present different applications of our results, including uniqueness of positive solutions for Lane-Emden systems in the subcritical case in a ball, and symmetry for a class of systems with natural growth in the gradient.

Remarks on infinitely many solutions for a class of Schr\xf6dinger equations with sign-changing potential

In this paper, we study the existence of infinitely many nontrivial solutions for the following semilinear Schrödinger equation: {−Δu+V(x)u=f(x,u),x∈RN,u∈H1(RN),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\textstyle\\begin{cases} -\\Delta u+V(x)u=f(x,u), \\quad x\\in\\mathbb{R}^{N},\\\\ u\\in H^{1}(\\mathbb{R}^{N}), \\end{cases} $$\\end{document} where the potential V is continuous and is allowed to be sign-changing. By using a variant fountain theorem, we obtain the existence of infinitely many high energy solutions under the condition that the nonlinearity f(x,u) is of super-linear growth at infinity. The super-quadratic growth condition imposed on F(x,u)=int _{0}^{u}f(x,t),dt is weaker than the Ambrosetti–Rabinowitz type condition and the similar conditions employed in the references.

Solitary waves in dispersive evolution equations of Whitham type with nonlinearities of mild regularity

We show existence of small solitary and periodic traveling-wave solutions in Sobolev spaces , , to a class of nonlinear, dispersive evolution equations of the formwhere the dispersion is a negative-order Fourier multiplier whose symbol is of KdV type at low frequencies and has integrable Fourier inverse and the nonlinearity is inhomogeneous, locally Lipschitz and of superlinear growth at the origin. This generalises earlier work by Ehrnström, Groves and Wahlén on a class of equations which includes Whitham’s model equation for surface gravity water waves featuring the exact linear dispersion relation. Tools involve constrained variational methods, Lions’ concentration-compactness principle, a strong fractional chain rule for composition operators of low relative regularity, and a cut-off argument for which enables us to go below the typical regime. We also demonstrate that these solutions are either waves of elevation or waves of depression when is nonnegative, and provide a nonexistence result when is too strong.

Truncated Euler-Maruyama method for classical and time-changed non-autonomous stochastic differential equations

The truncated Euler-Maruyama (EM) method is proposed to approximate a class of non-autonomous stochastic differential equations (SDEs) with the Hölder continuity in the temporal variable and the super-linear growth in the state variable. The strong convergence with the convergence rate is proved. Moreover, the strong convergence of the truncated EM method for a class of highly non-linear time-changed SDEs is studied.

Explicit Milstein schemes with truncation for nonlinear stochastic differential equations: Convergence and its rate

Although some implicit numerical procedures have been developed to treat high nonlinearity, the question whether one can use explicit schemes to achieve convergence rate similar to that of Milstein’s procedure remained open. This brings us to the current work that focuses on numerical solutions of stochastic differential equations using explicit schemes. Our main goals are to obtain order one convergence in the second moment in a finite-time interval. In contrast to the implicit schemes, explicit schemes are advantageous, easily implementable, and computationally less intensive. To overcome the difficulties due to super-linear growth of the coefficients, a truncation device is used in our algorithm. In addition to reaching aforementioned goals in the analysis part, numerical examples are provided to demonstrate our results.

Multiplicity of clines for systems of indefinite differential equations arising from a multilocus population genetics model

We investigate sufficient conditions for the presence of coexistence states for different genotypes in a diploid diallelic population with dominance distributed on a heterogeneous habitat, considering also the interaction between genes at multiple loci. In mathematical terms, this corresponds to the study of the Neumann boundary value problem p1′′+λ1w1(x,p2)f1(p1)=0,in Ω,p2′′+λ2w2(x,p1)f2(p2)=0,in Ω,p1′=p2′=0,on ∂Ω,where the coupling-weights wi are sign-changing in the first variable, and the nonlinearities fi:[0,1]→[0,+∞[ satisfy fi(0)=fi(1)=0, fi(s)>0 for all s∈]0,1[, and a superlinear growth condition at zero. Using a topological degree approach, we prove existence of 2N positive fully nontrivial solutions when the real positive parameters λ1 and λ2 are sufficiently large.

Positive Solutions of Fourth-Order Problem Subject to Nonlocal Boundary Conditions

In this paper, we study the fourth-order problem with the second derivative in nonlinearity under nonlocal boundary value conditions involving Stieltjes integrals. Some inequality conditions on nonlinearity and the spectral radius conditions of linear operators are presented that guarantee the existence of positive solutions to the problem by the theory of fixed point index on special cone. The conditions allow that the nonlinearity has superlinear or sublinear growth. Two examples are provided to support the main results under mixed boundary conditions involving multi-point with sign-changing coefficients and integral with sign-changing kernel.