Existence Of Nontrivial Solutions Research Articles

On solvability of one class of nonlinear integral equations on whole line with a weak singularity at zero

In present work we investigate one class of nonlinear integral equations with singularity at zero and boundary value conditions at ±∞. Above mentioned class of equations has direct applications in string theory and in the case of concrete structure of the kernel it describes the dynamics of the open-closed p-adic string for the scalar tachyon field. We prove the existence of nontrivial solution in a certain weight class of functions.With an additional restriction on the kernel the uniqueness of the obtained solution is proved.

Existence of nontrivial solutions for superlinear three-point boundary value problems

In this paper, we investigate the existence of nontrivial solutions for some superlinear second order three-point boundary value problems by applying new fixed point theorems in ordered Banach spaces with the lattice structure derived by Sun and Liu.

Existence and concentration of solutions for the nonlinear Kirchhoff type equations with steep well potential

In this paper, we study the following nonlinear problem of Kirchhoff type: \t\t\t{−(a+b∫R3|∇u|2)Δu+λV(x)u=|u|p−2u,in R3,u∈H1(R3),\\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}$$ \\textstyle\\begin{cases} - ( a + b\\int_{{\\mathbb{R}^{3}}} \\vert {\\nabla u} \\vert ^{2} ) \\Delta u + \\lambda V(x)u = {{ \\vert u \\vert } ^{p - 2}}u,\\quad \\text{in }{\\mathbb{R}^{3}}, \\\\ {u \\in{H^{1}}({\\mathbb{R}^{3}}),} \\end{cases} $$\\end{document} where the parameter lambda > 0 and 4 leq p < 6, constants a, b>0. By variational methods, the results of the existence of nontrivial solutions and the concentration phenomena of the solutions as lambda to + infty are obtained. It is worth pointing out that, for the case pin(4,6), the potential V is permitted to be sign-changing.

Existence of solutions for asymptotically periodic fractional Schrödinger equation

In this paper we study the following nonlinear fractional Schrödinger equation (−Δ)su+V(x)u=f(x,u),x∈RN,where s∈(0,1), (−Δ)s is the fractional Laplacian operator of order s, and V, f are asymptotically periodic functions in x. Under suitable conditions, the existence of nontrivial solution is obtained by using variational methods and concentration compactness argument.

Conditions for mode change of a rigid cavity

Initial conditions for the existence of an earlier observed change in the mode of radiation of a nonuniformly moving linear cavity with fixed contents are considered. It is shown that the imaginary phase of radiation is independent of the homogeneous medium that fills an accelerated cavity with a fixed phase structure. From the law of conservation of energy in a uniform motion, it is obtained in the operator form that the radiation phase is invariant to the constant velocity of motion of a rigid cavity. The necessary and sufficient conditions for the existence of nontrivial solutions of the equation of state of radiation are discussed. The synchronism condition for the interfering branches of radiation by a nonuniformly moving source is found. It is established that, for two fields with complex phases, the interference term exceeds eight-fold the intensity of the initial beam.

Existence results to a nonlinear p(k)-Laplacian difference equation

In the present paper, by using variational method, the existence of non-trivial solutions to an anisotropic discrete non-linear problem involving p(k)-Laplacian operator with Dirichlet boundary condition is investigated. The main technical tools applied here are the two local minimum theorems for differentiable functionals given by Bonanno.

A critical Kirchhoff type problem involving the fractional Laplacian in

In this paper, we are concerned with the existence of solutions for a critical p-Kirchhoff type problem driven by a nonlocal integro-differential operator:where is a continuous function, is a singular kernel function, is a nonlocal fractional operator, with , , f is a Carathéodory function on satisfying the Ambrosetti–Rabinowitz type condition. Under some suitable assumptions, we obtain the existence of nontrivial solutions for above problem by applying the mountain pass theorem. A distinguished feature of this paper is that M(0) may be zero, which means that the problem is degenerate. Consequently, the main theorem extends in several directions the recent results of Autuori, Fiscella and Pucci [Nonlinear Anal. 2015;125:699–714].

Elliptic equations in dimension 2 with double exponential nonlinearities

A boundary value problem on the unit disk in \(\mathbb R^2\) is considered, involving an elliptic operator with a singular weight of logarithmic type and nonlinearities which are subcritical or critical with respect to the associated gradient norm. The existence of non-trivial solutions is proved, relying on variational methods. In the critical case, the associated energy functional is non-compact. A suitable asymptotic condition allows to avoid the non-compactness levels of the functional.

Existence and uniqueness of solutions for a first-order discrete fractional boundary value problem

In this paper we use topological degree to establish the existence of nontrivial solutions for a first-order discrete fractional boundary value problem with a sign-changing nonlinearity. Also using the monotone iterative technique we discuss problems with a nonnegative nonlinearity and establish the uniqueness of positive solutions.

The Fu\u010dik spectrum of the p-Laplace equations with different weights and its resonance problems

In the present paper, our purpose is to construct the Fučik spectrum C_{l}^{pm} (cf. (2.13) and (2.17) below) with different weights for the p-Laplacian. As an application, we will discuss the existence of nontrivial solutions to the p-Laplace equations with resonance on the Fučik spectrum by making use of variational methods and Morse theory.

Schwinger mechanism in linear covariant gauges

In this work we explore the applicability of a special gluon mass generating mechanism in the context of the linear covariant gauges. In particular, the implementation of the Schwinger mechanism in pure Yang-Mills theories hinges crucially on the inclusion of massless bound-state excitations in the fundamental nonperturbative vertices of the theory. The dynamical formation of such excitations is controlled by a homogeneous linear Bethe-Salpeter equation, whose nontrivial solutions have been studied only in the Landau gauge. Here, the form of this integral equation is derived for general values of the gauge-fixing parameter, under a number of simplifying assumptions that reduce the degree of technical complexity. The kernel of this equation consists of fully-dressed gluon propagators, for which recent lattice data are used as input, and of three-gluon vertices dressed by a single form factor, which is modelled by means of certain physically motivated Ans\"atze. The gauge-dependent terms contributing to this kernel impose considerable restrictions on the infrared behavior of the vertex form factor; specifically, only infrared finite Ans\"atze are compatible with the existence of nontrivial solutions. When such Ans\"atze are employed, the numerical study of the integral equation reveals a continuity in the type of solutions as one varies the gauge-fixing parameter, indicating a smooth departure from the Landau gauge. Instead, the logarithmically divergent form factor displaying the characteristic "zero crossing", while perfectly consistent in the Landau gauge, has to undergo a dramatic qualitative transformation away from it, in order to yield acceptable solutions. The possible implications of these results are briefly discussed.

Existence of Solutions for a Fractional and Non-Local Elliptic Operator

In this paper, we consider a fractional and p-laplacian elliptic equation. In order to study this problem, we apply the technique of Nehari manifold and fibering map, which permit treating the existence of nontrivial solutions of a fractional and nonlocal equation, satisfies the homogeneous Dirichlet boundary conditions.

Nontrivial Solution for the Fractional p-Laplacian Equations via Perturbation Methods

We study the existence of nontrivial solution of the following equation without compactness: (-Δ)pαu+up-2u=f(x,u), x∈RN, where N,p≥2, α∈(0,1), (-Δ)pα is the fractional p-Laplacian, and the subcritical p-superlinear term f∈C(RN×R) is 1-periodic in xi for i=1,2,…,N. Our main difficulty is that the weak limit of (PS) sequence is not always the weak solution of fractional p-Laplacian type equation. To overcome this difficulty, by adding coercive potential term and using mountain pass theorem, we get the weak solution uλ of perturbation equations. And we prove that uλ→u as λ→0. Finally, by using vanishing lemma and periodic condition, we get that u is a nontrivial solution of fractional p-Laplacian equation.

High energy solutions of modified quasilinear fourth-order elliptic equations with sign-changing potential

In the present paper, we consider the following modified quasilinear fourth-order elliptic equation (1.1){Δ2u−Δu+V(x)u−12Δ(u2)u=f(x,u),forx∈RN,u(x)∈H2(RN), where Δ2:=Δ(Δ) is the biharmonic operator, V∈C(RN,R) and f∈C(RN×R,R), N≤5, are allowed to be sign-changing. Two main theorems on the existence of nontrivial solutions and infinitely many high energy solutions for Eq. (1.1) are obtained via variational methods.

Algebraic dependency of roots of multivariate polynomials and its applications to linear functional equations

In this paper we prove that a multivariate polynomial has algebraically dependent roots if and only if the coefficients are algebraic numbers up to a common proportional term; for the problem see section 4.4 in Varga-Vincze (On the characteristic polynomials of linear functional equations, Period Math Hungar 71(2):250–260, 2015). The case of univariate polynomials belongs to basic algebra. As far as we know the case of multivariate polynomials is not discussed in the literature. As an application we formulate a sufficient and necessary condition for the existence of non-trivial solutions of special types of linear functional equations. The criteria is based only on the algebraic properties of the parameters in the functional equation.

Conditions of dynamic change in radiation mode

The inhomogeneous wave equation for determining the previously experimentally detected change in the radiation phase in an accelerated cavity with stationary content is studied. It is shown that the radiation phase is independent of a homogeneous medium filling the accelerated cavity with unchanged phase structure. The necessary and sufficient conditions of the existence of nontrivial solutions of the equation are discussed.

Lyapunov’s type inequalities for hybrid fractional differential equations

We investigate new results about Lyapunov-type inequalities by considering hybrid fractional boundary value problems. We give necessary conditions for the existence of nontrivial solutions for a class of hybrid boundary value problems involving Riemann-Liouville fractional derivative of order $2<\alpha\le3$ . The investigation is based on a construction of Green’s functions and on finding its corresponding maximum value. In order to illustrate the results, we provide numerical examples.

On superlinear fractional advection dispersion equation in R N $\mathbb{R}^{N}$

We consider the fractional advection dispersion equation in $\mathbb {R}^{N}$ . The nonlinearity is superlinear but does not satisfy the Ambrosetti-Rabinowitz type condition. We obtain the existence of nontrivial solutions of the equations, improving a recent result of Zhang-Sun-Li (Appl. Math. Model. 38:4062-4075, 2014).

Multiple solutions for a nonlinear Schrödinger–Poisson system with sign-changing potential

This paper deals with the existence and multiplicity of nontrivial solutions for a nonlinear Schrödinger–Poisson system. Under some suitable conditions, some criteria on the existence of nontrivial solutions, including a ground state solution, two solutions and a sequence of nontrivial solutions {uk} with ‖uk‖→0 as k→+∞, is established by applying the Morse theory and variational methods. Some known results in the literatures are extended and improved.

Standing waves for a class of quasilinear Schrödinger equations

We study the existence of standing wave solutions for a class of quasilinear Schrödinger equations with subcritical or critical growthwhere is a given potential, is a parameter. By variational methods, we investigate the range of parameters where the existence of non-trivial solutions can be guaranteed. Especially, by combing Resonance and Hahn–Banach theorems, we improve some previous results on .