Nontrivial Solutions Research Articles

On class of bi-nonlocal problems involving fractional p(.,.)-Laplacian operators

This work aims to obtain the existence and multiplicity of solutions for a bi-nonlocal problem involving the p(.,.)-Laplacian operator. Under specific assumptions, the paper uses the Mountain Pass Theorem combined with the Ekeland variational approach to establish the existence of two nontrivial solutions for the problem. The existence of infinitely many solutions is proved using the Symmetric Mountain Pass Theorem and the Ljusternik-Schnirelman Theorem. Finally, we study a p(.,.)-Laplacian problem with non-variational form.

A universe from a Lagrangian fixed point

In this paper, we investigate the theoretical possibility that a Lagrangian fixed point, when applied to cosmological models, can drive dynamical evolution towards a bouncing universe. We analyze the physics of a Lagrangian fixed point within the context of a gravitational average effective action featuring scale-dependent couplings. To explore this concept, we develop a toy model set in a four-dimensional, spatially flat spacetime, anchored by a Lagrangian fixed point. Solving the cosmological equations of this model analytically, we identify several non-trivial solution branches. These branches are characterized by a modified scale factor and dynamic gravitational couplings, offering new insights into the behavior of cosmological models under these conditions.

Semiclassical solutions for fractional logarithmic Schrödinger equations with potentials unbounded below

In this paper, we consider the following fractional logarithmic Schrödinger equation \begin{equation*} \varepsilon^{2s}(-\Delta)^s u + V(x)u=u\log u^2\quad \text{in } \R^N, \end{equation*} where $\varepsilon> 0$, $N\ge 1$ and $V(x)\in C\big(\R^N,\R\big)$ is a potential which can be unbounded below at infinity. By considering a new penalization, we show that the problem has a nontrivial solution $u_{\varepsilon}$ concentrating at a local minimum of $V$ as $\varepsilon\to 0$.

Soliton solutions of nonlinear coupled Davey–Stewartson Fokas system using modified auxiliary equation method and extended (G′/G2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(G'/G^{2})$$\\end{document}-expansion method

The captivating realm of the nonlinear coupled Davey–Stewartson Fokas system is explored in this research paper. As a powerful tool, the proposed system is utilized for the realistic representation of various non-linear dynamical mechanisms in different fields of sciences and engineering including non-linear optical fibers, plasma physics and water waves theory. Two distinct exact methods, namely the modified auxiliary equation method and the extended (G′/G2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(G'/G^{2})$$\\end{document}-expansion method, are utilized to acquire the exact soliton solutions of the non-linear coupled Davey–Stewartson Fokas system. A plethora of novel soliton solutions containing anti-kink, kink, bright, dark, dark-bright, bright-dark and some other singular soliton solutions, have been obtained using the employed exact methods. The significance of proposed manuscript lies in the novelty of obtained solutions. Kink, dark and bright solitons have wide applications in optical fiber communications, plasma physics and water waves dynamics. The acquired nontrivial exact solutions contain exponential, trigonometric, rational and hyperbolic functions. Some obtained solutions are visually represented through graphical simulations of 3D, 2D-contour and 2D-line plots, providing a comprehensive visualization of the soliton dynamics.The modulation instability of the proposed nonlinear system has been investigated, which ensures the stability of the system.

Solvability of a sixth‐order boundary value problem with multi‐point and multi‐term integral boundary conditions

This paper aims to investigate the existence and uniqueness of solutions for a sixth‐order differential equation involving nonlocal and integral boundary conditions. Firstly, we obtain the properties of the relevant Green's functions. The existence result of at least one nontrivial solution is obtained by applying the Krasnoselskii–Zabreiko fixed point theorem. Moreover, we also establish the existence of unique solution to the considered problem via Hölder and Minkowski inequalities and Rus's theorem. Finally, two numerical examples are included to show the applicability of our main results.

Infinitely of solutions for fractional κ(ξ)$$ \kappa \left(\xi \right) $$‐Kirchhoff equation in Hκ(ξ)ϖ,ν;μ(Λ)$$ {\mathcal{H}}_{\kappa \left(\xi \right)}^{\varpi, \nu; \mu}\left(\Lambda \right) $$

This work aims to develop the variational framework for some Kirchhoff problems involving the ‐Hilfer operator. Precisely, we use the symmetric mountain pass theorem to prove the existence of unfairly of nontrivial solutions. Further, we research the results from the theory of variable exponent Sobolev spaces and from the theory of ‐fractional space .

Radiation of the energy-critical wave equation with compact support

We prove exterior energy lower bounds for (nonradial) solutions to the energy-critical nonlinear wave equation in space dimensions 3≤d≤5, with compactly supported initial data. In particular, it is shown that nontrivial global solutions with compact spatial support must be radiative in a sharp sense. In space dimensions 3 and 4, a nontrivial soliton background is also considered. As an application, we obtain partial results on the rigidity conjecture concerning solutions with the compactness property, including a new proof for the global existence of such solutions.

Minimal wave speed and spreading speed in predator-prey systems with stage structure

This article studies the minimal wave speed of traveling wave solutions as well as spreading speed in predator-prey systems. The model divides the prey into the immaturity and maturity, and the predator feeds on the immature individuals. For the traveling wave solutions connecting the predator-free equilibrium with the coexistence state, we consider two different diffusion modes. For both modes, the minimal wave speed is shown by presenting the existence or nonexistence of nontrivial traveling wave solutions for all wave parameters. When the diffusion is local, the minimal wave speed is the spreading speed of the corresponding initial value problem, in which the initial condition implies that the predator is the invader while the prey is the aborigine in the whole habitat. Under the persistence assumption in the corresponding kinetic systems, our conclusion coincides with the fact that the diffusion of the prey may have less effect on the invasion ability of the predator.

Anisotropic (p, q) Equation with Partially Concave Terms

We consider a Dirichlet problem driven by the anisotropic (p,q) Laplacian. In the reaction, we have a parametric partially concave term plus a “superlinear” perturbation (convex term) which need not satisfy the Ambrosetti–Rabinowitz condition. Using variational tools, we show that for all small values of the parameter λ>0, the problem has at least two nontrivial smooth solutions.

Propagation dynamics of a nonlocal dispersal Zika transmission model with general incidence

In this paper, we are interested in propagation dynamics of a nonlocal dispersal Zika transmission model with general incidence. When the threshold is greater than one, we prove that there is a wave speed such that the model has a traveling wave solution with speed , and there is no traveling wave solution with speed less than . When the threshold is less than or equal to one, we show that there is no nontrivial traveling wave solution. The approaches we use here are Schauder's fixed point theorem with an explicit construction of a pair of upper and lower solutions, the contradictory approach, and the two‐sided Laplace transform.

Solvability of a Hadamard fractional boundary value problem with multi‐term integral and Hadamard fractional derivative boundary conditions

In the present paper, we construct the existence of nontrivial solutions to a new kind of Hadamard fractional boundary value problem on an unbounded domain. With the contribution of some fixed point theorems in cone and the corresponding Green function, we ensure sufficient conditions for the Hadamard fractional boundary value problem. Also, the paper concludes with two examples to demonstrate our results.

Existence of solutions for critical Neumann problem with superlinear perturbation in the half‐space

AbstractIn this paper, we consider the existence and multiplicity of solutions for the critical Neumann problem where , , , , , is the outward unit normal vector at the boundary , is the usual critical exponent for the Sobolev embedding and is the critical exponent for the Sobolev trace embedding . By establishing an improved Pohozaev identity, we show that problem () has no nontrivial solution if . Applying the mountain pass theorem without the condition and the delicate estimates for the mountain pass level, we obtain the existence of a positive solution for all and the different values of the parameters and . Particularly, for , , , we prove that problem () has a positive solution if and only if . Moreover, the existence of multiple solutions for () is also obtained by dual variational principle for all and suitable .

On the size of finite Sidon sets

UDC 519.1 A Sidon set (also called a Golomb ruler) is a B 2 sequence and a 1 -thin set is a set of integers containing no nontrivial solutions to the equation a + b = c + d . We improve the lower bound for the diameter of a Sidon set with k elements, namely, if k is sufficiently large and 𝒜 is a Sidon set with k elements, then ⅆ i a m ( 𝒜 ) ≥ k 2 - 1.99405 k 3 / 2 . Alternatively, if n is sufficiently large, then the cardinality of the largest subset of { 1,2 , … , n } , which is a Sidon set, does not exceed n 1 / 2 + 0.99703 n 1 / 4 .

Nontrivial solutions for multiparameter k-Hessian systems

This paper is devoted to studying the coupled system of multiparameter k-Hessian equations where B = {x ∈ ℝ n : |x| < 1}, λ1, λ2 are positive parameters, ρi (i = 1, 2) is singular near the boundary ∂B, f i(i = 1, 2) satisfies a combined superlinear condition at ∞. Employing the fixed point theorem of Krasnosel’skii type in Banach space, the existence and multiplicity of nontrivial radial solutions are established. In particular, the dependence of the solutions on the parameters λ1, λ2 is also discussed. Finally, we study the nonexistence results of nontrivial radial solutions for the case λ1, λ2 ≫ 1.

Эксперименты с формой: варианты деконструкции и неоконструктивизма в архитектуре Москвы 1990-2000-х годов

At the turn of the 20th — 21st centuries, Moscow architecture was dynamic, with different architects working simultaneously, embodying individual meanings in a variety of architectural form and materials. One of the lines of the modern Moscow architecture development was experimentation with space, form and construction, which, among other things, involved references to the principles and techniques of deconstructivism and appeals to constructivism, which make the subject of this article. Deconstructivist motifs in the Moscow architecture of the 1990s — 2000s are primarily visible in the architectural works of A. Asadov’s studio and individual projects of V. Kolosnitsyn and A. Loktev. The search for non-trivial plastic solutions interested such architects as M. Khazanov, A. Bokov, S. Skuratov, S. Kiselev, G. Nadtochiy, K. Konovaltsev, B. Shabunin and others. Some of the architectural objects appeared to be full-fledged followers of constructivist traditions (e.g. the works of the architectural workshop of N. Lyzlov, some projects of A. Skokan and the Ostozhenka architectural bureau, urban objects of A. Vissarionov, A. Vorontsov, Y. Ilyin-Adaev, etc.). This article reveals one of the trends in the development of Moscow architecture which along with neomodernism, neoclassicism, capitalist romanticism (or ‘caprom’), an appeal to Art Deco, to the techniques and principles of postmodernism has permanently changed the appearance of the capital.

VIBRATIONS OF A VERTICAL BEAM ROTATING WITH VARIABLE ANGULAR VELOCITY

An Euler-Bernoulli beam in vertical position rotating about its symmetry axis along its length is considered. The angular velocity is assumed to have small fluctuations about a constant mean velocity. The partial differential equation of motion is derived first. The equation is cast into a non-dimensional form. The natural frequencies are calculated for the pinned-pinned case. Principle parametric resonances such that the fluctuation frequency being close to two times one of the natural frequencies are considered. By employment of the Method of Multiple Scales, an approximate perturbation solution is found. The frequency response diagrams are drawn and the bifurcation points for transition from the trivial solution to the non-trivial solution are calculated. The conditions for which such resonances occur are exploited in the numerical results.

Existence of nontrivial solutions for fractional Klein–Gordon equation coupled with Born–Infeld theory with critical exponents

In this paper, we investigate the existence of solutions for a fractional Klein–Gordon equation coupled with Born–Infeld theory when the nonlinearity exhibits critical growth. By overcoming the lack of compactness, we obtain that there is a radially symmetric solution by using variational methods.

Multiplicity and concentration of nontrivial solutions for Kirchhoff–Schrödinger–Poisson system with steep potential well

Multiplicity and concentration of nontrivial solutions for Kirchhoff–Schrödinger–Poisson system with steep potential well

Symmetry analysis, optimal system, conservation laws and exact solutions of time-fractional diffusion-type equation

In Liu et al. [On the generalized time fractional diffusion equation: Symmetry analysis, conservation laws, optimal system and exact solutions, Int. J. Geom. Methods Mod. Phys. 17(01) (2020) 2050013], the generalized time-fractional diffusion equation [Formula: see text] is studied by the symmetry analysis method. Liu et al. obtained two group generators [Formula: see text] [Formula: see text] and only one trivial solution [Formula: see text] for the equation. In this paper, we classify the Lie symmetry group admitted by the equation, and for the case [Formula: see text], we obtain a richer set of group generators and some nontrivial solutions, including unprecedented exact solutions and power series solutions. In addition, we construct the one-dimensional optimal system of the Lie symmetry group admitted by the time-fractional diffusion-type equation by Olver’s method, and obtain the conservation laws for all the obtained Lie symmetries using the general method developed by Ibragimov. For the novel exact solutions and power series solutions, we analyze their dynamic behavior graphically.

Existence and multiplicity of solutions for a p(x)-Choquard-Kirchhoff problem involving critical growth and concave-convex nonlinearities

This paper is devoted to studying a kind of p(x)-Choquard-Kirchhoff problems involving critical growth and concave-convex nonlinearities. Combining the concentration-compactness principle for weighted variable exponent spaces, the calculus of variations, genus theory and the Hardy-Littlewood-Sobolev type inequality, we obtain the existence of nontrivial solutions for this kind of p(x)-Choquard-Kirchhoff problems. Our results improve the related ones in the literature.