Kirchhoff Equations Research Articles

Normalized ground states for a kind of Choquard–Kirchhoff equations with critical nonlinearities

In this paper, we consider the existence of a normalized ground-state solution for the Choquard–Kirchhoff equation: {−(a+b∫R3|∇u|2dx)Δu=λu+μ(Iα∗|u|p)|u|p−2u+ω|u|4u,inR3,u>0,∫R3|u|2=m2,inR3,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\left \\{ \ extstyle\\begin{array}{l@{\\quad}r} -(a+b\\int _{\\mathbb{R}^{3}}|\ abla u|^{2}dx)\\Delta{u}=\\lambda u+\\mu (I_{ \\alpha}\\ast |u|^{p})|u|^{p-2}u+\\omega |u|^{4}u,\\ \\ &\ ext{in}\\, \\mathbb{R}^{3}, \\\\ u>0,\\quad \\int _{\\mathbb{R}^{3}}|u|^{2}=m^{2},\\,&\ ext{in}\\, \\mathbb{R}^{3}, \\end{array}\\displaystyle \\right . $$\\end{document} where a, b, m, μ, ω>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\omega >0$\\end{document}, p∈(2,7+α3)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$p\\in (2,\\frac{7+\\alpha}{3})$\\end{document}, λ∈R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\lambda \\in \\mathbb{R}$\\end{document}, α∈(0,3)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\alpha \\in (0,3)$\\end{document}, and Iα\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$I_{\\alpha}$\\end{document} is a Riesz potential. Utilizing approximation methods and Schwartz symmetrization rearrangements, we establish the existence of normalized ground states for these kinds of mass-constrained Choquard–Kirchhoff problems.

Bound states for fractional Kirchhoff equations with critical growth

This paper deals with the following fractional Kirchhoff problems: ( ε 2 s a + ε 4 s − N b ∫ R N | ( − Δ ) s 2 u | 2 d x ) ( − Δ ) s u + V ( x ) u = | u | 2 s ∗ − 2 u , in R N , where a, b>0, s ∈ ( 0 , 1 ) , 2s<N<4s, ε is a positive parameter, 2 s ∗ = 2 N N − 2 s is the critical Sobolev exponent, V ( x ) ∈ L N 2 s ( R N ) and V ( x ) vanishes at some points in R N . In virtue of a global compactness result and Ljusternik–Schnirelman theory, we succeed in proving the multiplicity of bound states.

Advanced Modeling of Hydrogen Turbines Using Generalized Conformable Calculus

This article addresses critical challenges in the transition to clean energy sources by highlighting the importance of advanced mathematical modeling and computational techniques in turbine design and operation. Specifically, we extend and generalize the work of Camporeale to advance the modeling of hydrogen turbine systems. By utilizing conformable calculus, we develop dynamic equations that analyze key aspects of turbine performance, including temperature variations in turbine blades, angular velocities of rotating shafts, and mass--energy balances within the plenum and combustion chamber. Furthermore, we incorporate Kirchhoff's equation in its generalized conformable integral form, enhancing the precision of energy balance calculations and improving the representation of heat transfer processes in the combustion chamber. This methodology introduces novel perspectives in hydrogen turbine research, contributing to the advancement of sustainable and efficient technologies. Our comprehensive approach aims to provide more accurate and efficient predictions of turbine behavior, thereby impacting the design and optimization of hydrogen-based clean energy systems.

Normalized Solutions for Kirchhoff Equations with Exponential Nonlinearity and Singular Weights

Normalized Solutions for Kirchhoff Equations with Exponential Nonlinearity and Singular Weights

Normalized solutions for the Kirchhoff equation with combined nonlinearities in ℝ4

Abstract In this article, we study the following Kirchhoff equation with combined nonlinearities: − a + b ∫ R 4 ∣ ∇ u ∣ 2 d x Δ u + λ u = μ ∣ u ∣ q − 2 u + ∣ u ∣ 2 u , in R 4 , ∫ R 4 ∣ u ∣ 2 d x = c 2 , \left\{\begin{array}{l}-\left(a+b\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{4}}{| \nabla u| }^{2}{\rm{d}}x\right)\Delta u+\lambda u=\mu {| u| }^{q-2}u+{| u| }^{2}u,\hspace{1.0em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{4},\\ \mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{4}}{| u| }^{2}{\rm{d}}x={c}^{2},\end{array}\right. where a , b , c &gt; 0 a,b,c\gt 0 , μ , λ ∈ R \mu ,\lambda \in {\mathbb{R}} , 2 &lt; q &lt; 4 2\lt q\lt 4 . Under different assumptions on b , c &gt; 0 b,c\gt 0 and μ ∈ R \mu \in {\mathbb{R}} , we prove some existence, nonexistence, and asymptotic behavior of the obtained normalized solutions. When μ &gt; 0 \mu \gt 0 :(i) for 2 &lt; q &lt; 3 2\lt q\lt 3 , we obtain the existence of a local minimizer ground-state solution and a mountain-pass-type solution, (ii) for q = 3 q=3 and 3 &lt; q &lt; 4 3\lt q\lt 4 , we obtain the existence of a mountain-pass type ground-state solution respectively, under different assumptions. When μ &lt; 0 \mu \lt 0 and 2 &lt; q &lt; 4 2\lt q\lt 4 , we prove the nonexistence result of the aforementioned problem. We also investigate the asymptotic behavior of the normalized ground-state solutions, when μ → 0 + \mu \to {0}^{+} and b → 0 + b\to {0}^{+} , respectively.

Multiplicity of solutions for a Kirchhoff equation with non‐linearity concave at the origin

We are concerned with the Kirchhoff problem where is a bounded domain with a regular boundary , and , with for and for . Given appropriate conditions on the function , we successfully applied variational methods to establish the existence of multiple solutions, including both a ground state and a nodal solution, for the specified problem, provided that is sufficiently small and for certain values of .

Existence and limit behavior of normalized ground state solutions for a class of non-autonomous Kirchhoff equations

Existence and limit behavior of normalized ground state solutions for a class of non-autonomous Kirchhoff equations

Long-time dynamics of a random higher-order Kirchhoff model with variable coefficient rotational inertia

This paper delves into the stochastic asymptotic behavior of a non-autonomous stochastic higher-order Kirchhoff equation with variable coefficient rotational inertia. The equation is solved using the Galerkin method, and a stochastic dynamical system is established on this basis. Uniform estimation demonstrates a family of Dk−absorbing sets in the stochastic dynamical system Φk, and the asymptotic compactness of Φk is proved via decomposition. Finally, the family of Dk−random attractors is acquired for the stochastic dynamical system Φk in Vm+k(Ω)×Vm+kb0(Ω). These results improve and extend those in recent literature (Lv et al., 2021). The findings promote the relevant conclusions of the non-autonomous stochastic higher-order Kirchhoff model and provide a theoretical basis for its subsequent application and research.

Normalized solutions for Kirchhoff equations with Sobolev critical exponent and mixed nonlinearities

Normalized solutions for Kirchhoff equations with Sobolev critical exponent and mixed nonlinearities

Geometrical description and Faddeev-Jackiw quantization of electrical networks

In lumped-element electrical circuit theory, the problem of solving Maxwell&amp;apos;s equations in the presence of media is reduced to two sets of equations, the constitutive equations encapsulating local geometry and dynamics of a confined energy density, and the Kirchhoff equations enforcing conservation of charge and energy in a larger, topological, scale. We develop a new geometric and systematic description of the dynamics of general lumped-element electrical circuits as first order differential equations, derivable from a Lagrangian and a Rayleigh dissipation function. Through the Faddeev-Jackiw method we identify and classify the singularities that arise in the search for Hamiltonian descriptions of general networks. The core of our solution relies on the correct identification of the reduced manifold in which the circuit state is expressible, e.g., a mix of flux and charge degrees of freedom, including the presence of compact ones. We apply our fully programmable method to obtain (canonically quantizable) Hamiltonian descriptions of nonlinear and nonreciprocal circuits which would be cumbersome/singular if pure node-flux or loop-charge variables were used as a starting configuration space. We also propose a specific assignment of topology for the branch variables of energetic elements, that when used as input to the procedure gives results consistent with classical descriptions as well as with spectra of more involved quantum circuits. This work unifies diverse existent geometrical pictures of electrical network theory, and will prove useful, for instance, to automatize the computation of exact Hamiltonian descriptions of superconducting quantum chips.

Asymptotic analysis of geometrically nonlinear beam vibrations: Kirchhoff and Bolotin equations

AbstractThe paper analyzes various approximate models of geometrically nonlinear vibrations of a beam. In practice, simplified equations are often based on the quasi‐static Kirchhoff hypothesis—neglecting axial inertia. This hypothesis is justified with the prescribed end‐displacements of the beam in the axial direction. Under dead loading, quasi‐static Kirchhoff hypothesis results in a linear equation. The corresponding approximate equations obtained in this paper are based on the asymptotic procedure. The ratio of bending stiffness to reduced tensile/compressive stiffness is taken as a small parameter. Axial inertia is taken into account in the equation of the first approximation. Introduced by V.V. Bolotin concept “nonlinear inertia” is discussed. The most common errors in using the quasi‐static Kirchhoff hypothesis are analyzed.

Global solvability for semidiscrete Kirchhoff equation

Global solvability for semidiscrete Kirchhoff equation

Blow-up and global existence for a new class of parabolic p(x,⋅)-Kirchhoff equation involving double phase operator

In this paper, we are concerned with study solutions for double phase problems involving a Kirchhoff function, and nonlinearity with subcritical growth:{ut+M([u]p,q,as)Lp,q,asu=|u|r−2u,inU×[0,T),u(x,t)=0in∂U×[0,T),u(x,0)=u0(x),inU, where, Lp,q,ss is double phase operator, and M is a Kirchhoff function. Combining the Faedo-Galerking approximation with Gronwall's inequality, the existence of a unique weak solution is established. More interestingly, we study the global existence and finite time blow of solution when the initial energy is sub-critical and supercritical. Finally, the stabilization of the solution with positive initial energy is established based on Komornik's integral inequality.

Existence and multiplicity of solutions for a new p(x)-Kirchhoff equation

Abstract This article is devoted to study a class of new p ( x ) p\left(x) -Kirchhoff equation. By means of perturbation technique, variational method, and the method invariant sets for the descending flow, the existence and multiplicity of solutions to this problem are obtained.

A critical Kirchhoff equation with a logarithmic type perturbation in dimension 4

In this paper, a critical Kirchhoff type elliptic equation involving a logarithmic type perturbation is considered in a bounded domain in . Three main features of the problem bring essential difficulties when proving the existence of weak solutions. The first one is the nonlocal term which makes the structure of the corresponding energy functional more complicated, the second one is that the problem is critical in the sense that the energy functional lacks compactness, and the third one is the appearance of the logarithmic term which satisfies neither the standard monotonicity condition nor the Ambrosetti–Rabinowitz condition. Moreover, the boundedness of the sequence is hard to obtain for Kirchhoff problems in dimension 4. By combining a result by Jeanjean (Proc. Roy. Soc. Edinburgh Sect. A, 1999, 129: 787–809) and a recent estimate by Deng et al. (Adv. Nonlinear Stud., 2023, 23: No. 20220049) with the mountain pass lemma and Brézis‐Lieb's lemma, it is proved that either the norm of the sequence of approximation solution goes to infinity or the problem admits a nontrivial weak solution, under appropriate assumptions on the parameters.

Infinitely Many Solutions for Schrödinger–Poisson Systems and Schrödinger–Kirchhoff Equations

By applying Clark’s theorem as altered by Liu and Wang and the truncation method, we obtain a sequence of solutions for a Schrödinger–Poisson system −Δu+V(x)u+ϕu=f(u)inR3,−Δϕ=u2inR3 with negative energy. A similar result is also obtained for the Schrödinger-Kirchhoff equation as follows:−1+∫RN∇u2Δu+V(x)u=f(u)u∈H1(RN).

Semilinear approximations of quasilinear parabolic equations with applications

An approach to solvability of certain quasilinear parabolic equations is presented by approximating the quasilinear equation under consideration with a parameter family of semilinear problems with stronger linear fractional diffusion term. Defined on arbitrarily long time intervals, solutions to the original problem are found as a suitable limit of global solutions to those semilinear approximations. The method is applied to nonlinear parabolic Kirchhoff equation, quasilinear reaction–diffusion equation, and critical 2D surface quasi‐geostrophic equation associated with Dirichlet boundary conditions.

Asymptotic behavior for Kirchhoff type stochastic plate equations on unbounded domains

This paper is concerned with the asymptotic behaviour of solutions to a class of Kirchhoff type stochastic nonlinear plate equations with dispersive and viscosity dissipative terms driven additive noise defined on the entire space Rn. The existence, uniqueness as well as upper semi-continuity of pullback random attractors are proved in H2(Rn)×H1(Rn).

Multi-bump solutions to Kirchhoff type equations in the plane with the steep potential well vanishing at infinity

In this paper, we study the following Kirchhoff type equation with a steep potential well vanishing at infinity:−(a+b∫R2|∇u|2dx)Δu+(h(x)+μV(x))u=K(x)f(u)inR2, where a,b>0 are constants, μ>0 is a parameter, V decays to zero at infinity as |x|−γ with γ∈(0,2] and intV−1(0) possesses multiple disjoint bounded components. We prove the existence of multi-bump solutions for μ>0 large enough and the concentration behavior of multi-bump solutions as μ→+∞.

Multiplicity and Concentration Behavior of Solutions to a Class of Fractional Kirchhoff Equation Involving Exponential Nonlinearity

Multiplicity and Concentration Behavior of Solutions to a Class of Fractional Kirchhoff Equation Involving Exponential Nonlinearity