Weak Growth Condition Research Articles

Existence results for singular nonlinear BVPs in the critical regime

We study the existence of solutions for a class of boundary value problems on the half line, associated to a third order ordinary differential equation of the type ( Φ ( k ( t , u ′ ( t ) ) u ″ ( t ) ) ) ′ ( t ) = f ( t , u ( t ) , u ′ ( t ) , u ″ ( t ) ) , a.a. t ∈ R 0 + . The prototype for the operator Φ is the Φ -Laplacian; the function k is assumed to be continuous and it may vanish in a subset of zero Lebesgue measure, so that the problem can be singular; finally, f is a Carathéodory function satisfying a weak growth condition of Winter–Nagumo type. The approach we follow is based on fixed point techniques combined with the upper and lower solutions method.

Concentrating solutions for singularly perturbed fractional (N/s)-Laplacian equations with nonlocal reaction

Abstract This paper is concerned with the following fractional (N/s)-Laplacian Choquard equation: ε N ⁢ ( - Δ ) N / s s ⁢ u + V ⁢ ( x ) ⁢ | u | N s - 2 ⁢ u = ε μ ⁢ ( 1 | x | N - μ ∗ F ⁢ ( u ) ) ⁢ f ⁢ ( u ) , x ∈ ℝ N , \varepsilon^{N}(-\Delta)_{N/s}^{s}u+V(x)\lvert u\rvert^{\frac{N}{s}-2}u=% \varepsilon^{\mu}\Bigl{(}\frac{1}{\lvert x\rvert^{N-\mu}}\ast F(u)\Bigr{)}f(u)% ,\quad x\in{\mathbb{R}}^{N}, where ( - Δ ) N / s s {(-\Delta)_{N/s}^{s}} denotes the (N/s)-Laplacian operator, 0 < μ < N {0<\mu<N} , and V and f are continuous real functions satisfying some mild assumptions. Applying the weak growth conditions on the exponential critical nonlinearity f and without using the strictly monotone condition, we use some refined analysis and develop the arguments in the existing results to establish the existence of the ground state solution of the fractional (N/s)-Laplacian Choquard equation. Moreover, we also study the concentration phenomenon of the ground state solutions. As far as we know, our results seem to be new concerning the fractional (N/s)-Laplacian equation with the Choquard reaction.

Existence and regularity of solutions for a class of fractional Laplacian problems

In this paper, we first establish a modified Marcinkiewicz interpolation result, and using this result, we obtain a new regularity result for a fractional Laplacian problem on a bounded C1,1-domain Ω in RN, and we also obtain a new regularity result for then fractional problem on unbounded Ω=RN by using the Stampacchia truncation method. Next, by the Leray-Schauder fixed point theorem we obtain the existence of solutions for a class of fractional Laplacian problems with weak growth conditions on the nonlinearities. Finally, as an application, we prove the existence of positive solutions for a fractional Laplacian problem which has an exponential growth nonlinearity.

Approximate controllability of fractional stochastic evolution equations with nonlocal conditions

Abstract This paper deals with the approximate controllability for a class of fractional stochastic evolution equations with nonlocal initial conditions in a Hilbert space. We delete the compactness condition or Lipschitz condition for nonlocal term appearing in various literatures, and only need to suppose some weak growth condition on the nonlocal term. The discussion is based on the fixed point theorem, diagonal argument and approximation techniques. In the end, an example is presented to illustrate the abstract theory.

Existence results for boundary value problems associated with singular strongly nonlinear equations

We consider a strongly nonlinear differential equation of the following general type: $$\begin{aligned} (\Phi (a(t,x(t)) \, x'(t)))'= f(t,x(t),x'(t)), \quad \text {a.e. on }[0,T], \end{aligned}$$where f is a Carathédory function, \(\Phi \) is a strictly increasing homeomorphism (the \(\Phi \)-Laplacian operator), and the function a is continuous and non-negative. We assume that a(t, x) is bounded from below by a non-negative function h(t), independent of x and such that \(1/h \in L^p(0,T)\) for some \(p> 1\), and we require a weak growth condition of Wintner–Nagumo type. Under these assumptions, we prove existence results for the Dirichlet problem associated with the above equation, as well as for different boundary conditions. Our approach combines fixed point techniques and the upper/lower solution method.

Standing waves with a critical frequency for a quasilinear Schrödinger equation

We consider the following quasilinear Schrödinger equation [Formula: see text] where [Formula: see text], [Formula: see text], and [Formula: see text] satisfies a weaker growth condition than the Ambrosetti–Rabinowitz type condition in Byeon and Wang [Standing waves with a critical frequency for nonlinear Schrödinger equations, Arch. Ration. Mech. Anal. 165(4) (2002) 295–316; Standing waves with a critical frequency for nonlinear Schrödinger equations, II, Calc. Var. 18(2) (2003) 207–219]. We obtain the existence of the localized bound state solutions concentrating at an isolated component of the local minimum of [Formula: see text] and whose amplitude goes to 0 as [Formula: see text].

Approximation Technique for Fractional Evolution Equations with Nonlocal Integral Conditions

We consider a class of fractional evolution equations with nonlocal integral conditions in Banach spaces. New existence of mild solutions to such a problem are established using Schauder fixed-point theorem, diagonal argument and approximation techniques under the hypotheses that the nonlinear term is Caratheodory continuous and satisfies some weak growth condition, the nonlocal term depends on all the value of independent variable on the whole interval and satisfies some weak growth condition. This work may be viewed as an attempt to develop a general existence theory for fractional evolution equations with general nonlocal integral conditions. Finally, as a sample of application, the results are applied to a fractional parabolic partial differential equation with nonlocal integral condition. The results obtained in this paper essentially extend some existing results in this area.

A classification of harmonic Maass forms

We give a classification of the Harish-Chandra modules generated by the pullback to $$\mathrm{SL}_2(\mathbb {R})$$ of harmonic Maass forms for congruence subgroups of $$\mathrm{SL}_2(\mathbb {Z})$$ with exponential growth allowed at the cusps. We assume that the weight is integral but include vector-valued forms. Due to the weak growth condition, these modules do not need to be irreducible. Elementary Lie algebra considerations imply that there are nine possibilities, and we show, by giving explicit examples, that all of them arise from harmonic Maass forms. Finally, we briefly discuss the case of forms that are not harmonic but rather are annihilated by a power of the Laplacian, where much more complicated Harish-Chandra modules can arise. We hope that our classification will prove useful in understanding harmonic Maass forms from a representation theoretic perspective and that it will illustrate in the simplest case the phenomenon of extensions occurring in the space of automorphic forms.

On the Convergence Rate of a Proximal Point Algorithm for Vector Function on Hadamard Manifolds

The proximal point algorithm has many interesting applications, such as signal recovery, signal processing and others. In recent years, the proximal point method has been extended to Riemannian manifolds. The main advantages of these extensions are that nonconvex problems in classic sense may become geodesic convex by introducing an appropriate Riemannian metric, constrained optimization problems may be seen as unconstrained ones. In this paper, we propose an inexact proximal point algorithm for geodesic convex vector function on Hadamard manifolds. Under the assumption that the objective function is coercive, the sequence generated by this algorithm converges to a Pareto critical point. When the objective function is coercive and strictly geodesic convex, the sequence generated by this algorithm converges to a Pareto optimal point. Furthermore, under the weaker growth condition, we prove that the inexact proximal point algorithm has linear/superlinear convergence rate.

A priori estimates and existence of positive solutions for higher-order elliptic equations

In this paper, we present a new sufficient condition to get a priori L∞-estimates for positive solutions of higher-order elliptic equations in a smooth bounded convex domain of RN with Navier boundary conditions or for radially symmetric solutions in the ball with Dirichlet boundary conditions. A priori L∞-estimates for positive solutions of the second-order elliptic system in a smooth bounded convex domain of RN with Dirichlet boundary conditions are also established. As usual, these a priori bounds allow us to obtain existence results. Also, by truncation technique combined with minimax method, we obtain existence of positive solution for higher-order elliptic equations of the form (1.1) below when we only assume that the nonlinearity is a nondecreasing positive function satisfying: liminfs→+∞f(s)s>Λ1, limsups→0f(s)s<Λ1, where Λ1 is the first eigenvalue of (−Δ)m with Navier boundary conditions and the weak subcritical growth condition: lims→+∞⁡f(s)sσ=0, where σ=N+2mN−2m.

Littlewood-Paley theory on metric spaces with non doubling measures and its applications

The purpose of this paper is to extend the Littlewood-Paley theory to a geometrically doubling metric space with a non-doubling measure satisfying a weak growth condition. Moreover, we prove that our setting mentioned above, is equivalent to the one introduced and studied by Hytonen (2010) in his remarkable framework, i.e., the geometrically doubling metric space with a non-doubling measure satisfying a so-called upper doubling condition. As an application, we obtain the T1 theorem in this more general setting. Moreover, the Gaussian measure is also discussed.

EXISTENCE OF SOLUTIONS FOR A CLASS OF $p$-LAPLACIAN SYSTEMS WITH IMPULSIVE EFFECTS

By using the least action principle and the saddle point theorem, some existence theorems are obtained for second-order $p$-Laplacian systems with or without impulsive effects under weak sublinear growth conditions, we improve some existing results in the literature.

Inequalities for differences of Dyson’s rank for all odd moduli

A partition of a non-negative integer n is any non-increasing sequence of positive integers whose sum is n. As usual, let p(n) denote the number of partitions of n. The partition function satisfies the famous “Ramanujan congruences” declaring that for c ∈ {5, 7, 11} we have for all n ≥ 0 that p(cn + δc) ≡ 0 (mod c), where δc is defined by the congruence 24δc ≡ 1 (mod c). In order to understand these from a combinatorial point of view, Dyson defined the rank of a partition as its largest part minus its number of parts [11]. Atkin and Swinnerton-Dyer [4] later proved that Dyson’s rank provides a combinatorial explanation of the congruences modulo 5 and 7, but not the congruence modulo 11. To simplify notation, for integers 0 ≤ a < c, we let N(a, c;n) to be the number of partitions of n whose rank is congruent to a (mod c). Rank differences have been the focus of several works and lead to interesting new automorphic forms, so called harmonic Maass forms. Harmonic Maass forms are generalizations of modular forms, in that they satisfy the same transformation law, and (weak) growth conditions at cusps, but instead of being holomorphic, they are annihilated by the weight k hyperbolic Laplacian. As an example, consider the function

Existence and uniqueness of solutions of stochastic functional differential equations

Using a variant of the Euler-Murayama scheme for stochastic functional dierential equations with bounded memory driven by Brownian motion we show that only weak one-sided local Lipschitz (or ’monotonicity’) conditions are sucient for local existence and uniqueness of strong solutions. In case of explosion the method yields the maximal solution up to the explosion time. We also provide a weak growth condition which prevents explosions to occur. In an appendix we formulate and prove four lemmas which may be of independent interest: three of them can be viewed as rather general stochas

On the Speed of Spread for Fractional Reaction‐Diffusion Equations

The fractional reaction diffusion equation ∂tu + Au = g(u) is discussed, where A is a fractional differential operator on ℝ of order α ∈ (0, 2), the C1 function g vanishes at ζ = 0 and ζ = 1, and either g ≥ 0 on (0, 1) or g &lt; 0 near ζ = 0. In the case of nonnegative g, it is shown that solutions with initial support on the positive half axis spread into the left half axis with unbounded speed if g(ζ) satisfies some weak growth condition near ζ = 0 in the case α &gt; 1, or if g is merely positive on a sufficiently large interval near ζ = 1 in the case α &lt; 1. On the other hand, it shown that solutions spread with finite speed if g′(0) &lt; 0. The proofs use comparison arguments and a suitable family of travelling wave solutions.

Periodic solutions for sublinear systems via variational approach

The variational method is used to obtain some existence theorems for periodic solutions of second order systems with or without impulsive under weak sublinear growth conditions.

Two-weight weak-type inequalities for potential type operators

For the potential type operator (The equation is abbreviated) where Φ is a non-negative locally integrable function on R(superscript n) and satisfies weak growth condition, a two-weight weak-type (p, q) inequality for T(subscript Φ) is obtained.

Convergence of Rational Interpolants to Analytic Functions with Restricted Growth

Let D be a regular domain in \(\overline{\rm C}\) with bounded boundary. For each n≥ 1, take points \(A_n=\{a_{ni}\}_{i=0}^n\) in D and points \(B_n=\{b_{ni}\}_{i=1}^n\) in \(\overline{\rm C}\), such that \tu Bn has no limit points in D. Let \gan and \gbn be the normalized point counting measures of An and Bn respectively and \(\alpha^\prime_n\) and \(\beta^\prime_n\) their swept measures onto \t6D. Denote by U\gm the logarithmic potential of the measure \gm. It has been shown by Ambroladze and Wallin, that if for every weak-star limit point of \gabn, we have \ga(D) 0 and also \(\lim_{n\to\infty}\bigg[\sup_{z\in\partial D}\bigg({n+1\over n}U_{\alpha^\prime_n}(z)-U_{\beta^\prime_n}(z)\bigg)\bigg]=0,\) then for every bounded analytic function f on D, the rational interpolants rn of degree n with poles at Bn interpolating to f at An, converge locally uniformly with geometric degree of convergence to f on D. We show that under a slightly stricter condition on {Bn}, the boundedness requirement on f can be replaced with the weaker growth condition \(\mid f(z)\mid \leq Ad(z,\partial D)^q,\) where A0, q< 0 and that this growth condition is also sharp in the sense described in this paper.

Uniqueness of non-harmonic series under a weaker growth condition

The uniqueness theorem for non-harmonic Fourier series is proved under a much weaker condition that for a number γ > 0 and a constant h > 0.

Weak sharp efficiency and growth condition for vector-valued functions with applications

We define weak sharp solutions to vector optimization problems and the growth condition for vector-valued functions. When applied to scalar-valued functions, weak sharp solutions reduce to weak sharp minima, and the growth condition reduces to the growth condition used in proving Hölder calmness of the solution set to parametric scalar optimization problems. By using these concepts We prove upper Hölder continuity and Hölder calmness of the solution set-valued mapping to parametric vector optimization problems.