Kirchhoff Type Equations Research Articles

Existence and asymptotic behavior of normalized solutions for fourth‐order equations of Kirchhoff type

In this paper, we consider the following fourth‐order equation of Kirchhoff type under the normalized constraint , where are constants, and is a Lagrange multiplier. Firstly, we obtain the existence of ground state solutions for or with . Moreover, we show the nonexistence of solutions for . Finally, we investigate the asymptotic behavior of normalized solutions obtained above with respect to the parameter .

Existence and multiplicity for fractional differential equations with $$m(\xi )$$-Kirchhoff type-equation

Existence and multiplicity for fractional differential equations with $$m(\xi )$$-Kirchhoff type-equation

Existence results for variable-order fractional Kirchhoff equations with variable exponents

This study establishes the existence of two nontrivial weak solutions for a specific class of Kirchhoff-type equations. Utilizing the Mountain Pass Theorem and the Ekeland variational principle, alongside insights from fractional Sobolev spaces with variable orders and exponents, we provide a concise demonstration of our result.

Concentration of Normalized Solutions for Mass Supercritical Kirchhoff Type Equations

Concentration of Normalized Solutions for Mass Supercritical Kirchhoff Type Equations

Existence of Nodal Solutions with Arbitrary Number of Nodes for Kirchhoff Type Equations

Existence of Nodal Solutions with Arbitrary Number of Nodes for Kirchhoff Type Equations

Nodal solutions for a nonlocal fourth order equation of Kirchhoff type

We study the bifurcation behavior of nodal solutions for the Kirchhoff type beam equation u′′′′−M(∫01|u′|2dx)u′′=λf(x,u),x∈(0,1),u(0)=u(1)=u′′(0)=u′′(1)=0, where λ∈R is a parameter, M:[0,∞)→[0,∞) and f:[0,1]×R→R are smooth functions. We obtain the existence of nodal solutions under some suitable conditions. The proof of our main result is based upon bifurcation techniques.

Normalized solutions for the general Kirchhoff type equations

Normalized solutions for the general Kirchhoff type equations

Optimal control problem governed by a kind of Kirchhoff-type equation

In this paper, we consider an optimal control problem governed by a kind of Kirchhoff-type equation, which plays an important role in the phenomenon of beam vibration. Firstly, the existence of solution to the state equation is proved by the variational method. Secondly, for the given cost functional, we get that there exists at least an optimal state-control pair via the Sobolev’s embedding theorem under the constraint of the state equation. Next, the necessary optimality condition for the optimal solution is derived by using the cone method. Finally, we give the pointwise variational inequality, minimum principles and an equivalent necessary condition for the optimal control problem according to the discussion of the variational inequality.

Limit Property of an L2-Normalized Solution for an L2-Subcritical Kirchhoff-Type Equation with a Variable Exponent

Limit Property of an L2-Normalized Solution for an L2-Subcritical Kirchhoff-Type Equation with a Variable Exponent

Well-posedness of nonautonomous abstract Cauchy problems under dissipativity conditions expressed by metric-like functionals

Under a dissipativity condition expressed by a metric-like functional, the well-posedness of a nonautonomous abstract Cauchy problem is established. The main result not only extends Martin's result on the generation of an evolution operator in a class of uniformly convex Banach spaces but also can be applied to mixed problems for nonautonomous evolution equations of Kirchhoff type with which Kato's quasilinear theory or the theory of quasi-contractive semigroups cannot directly deal. The main result also gives an affirmative answer to Kōmura's conjecture that an evolution operator of linear operators with moving domains can be generated even under a weak stability condition rather than Kato's stability condition, and the result provides an effective means for solving degenerate wave equations.

Infinitely many nodal solutions of Kirchhoff-type equations with asymptotically cubic nonlinearity without oddness hypothesis

Infinitely many nodal solutions of Kirchhoff-type equations with asymptotically cubic nonlinearity without oddness hypothesis

Existence and concentration behavior of normalized solutions for critical Kirchhoff type equations with general nonlinearities

Existence and concentration behavior of normalized solutions for critical Kirchhoff type equations with general nonlinearities

Existence results for a class of singular p(x)-Kirchhoff equations

In the present paper, we study a class of p ( x ) -Kirchhoff-type equations involving variable singularities. Using the Ekeland's variational principle, a constrained minimization and a global minimum method, we show the existence and uniqueness of positive solutions for the weak ( γ ∈ ( 0 , 1 ) ) and strong ( γ ∈ ( 1 , ∞ ) ) variable singularities. We provide two applications to illustrate the main results.

A Generalized Brezis–Lieb Lemma on Graphs and Its Application to Kirchhoff Type Equations

A Generalized Brezis–Lieb Lemma on Graphs and Its Application to Kirchhoff Type Equations

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→+∞.

Sign‐changing solution to a critical p‐Kirchhoff equation with potential vanishing at infinity in ℝN

In this paper, we study the critical ‐Laplacian equation of Kirchhoff type where is the Kirchhoff function, with and is a parameter, , and the potentials and vanish at infinity. Under some suitable assumptions on , by using the constraint minimization approach, we obtain a least energy sign‐changing solution to this problem if is large enough and show the energy of is strictly larger than twice that of the ground state solutions. Moreover, by considering a wider class of and , we exploit the truncation argument to find a nontrivial solution if is sufficiently large via some analytic skills.

Multi-bump solutions to Kirchhoff type equations with exponential critical growth in $$\mathbb {R}^2$$

Multi-bump solutions to Kirchhoff type equations with exponential critical growth in $$\mathbb {R}^2$$

Exact solutions and bifurcation curves of nonlocal elliptic equations with convolutional Kirchhoff functions

We study the one-dimensional nonlocal elliptic equation of Kirchhoff type with convolutional Kirchhoff functions. We establish the exact solutions uλ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$u_{\\lambda}$\\end{document} and bifurcation curves λ(α)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\lambda (\\alpha )$\\end{document}, where α:=∥uλ∥∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\alpha := \\Vert u_{\\lambda}\\Vert _{\\infty}$\\end{document}.

Multiplicity results for a Kirchhoff type equations with general potential

Multiplicity results for a Kirchhoff type equations with general potential

Multiplicity of solutions for Kirchhoff type equations with critical growth in ℝ N

In this paper, we consider the following Kirchhoff type problem with critical growth: { − ( a + b ∫ R N | ∇u | 2 d x ) Δu = λk ( x ) | u | q − 2 u + μ | u | 2 ∗ − 2 u , x ∈ R N , u ∈ D 1 , 2 ( R N ) , where N ≥ 3 , a ≥ 0 , b>0, 1<q<2, λ , μ > 0 , 2 ∗ = 2 N N − 2 is the critical Sobolev exponent and k ( x ) ∈ L 2 ∗ / ( 2 ∗ − q ) ( R N ) is a positive function. Using variational methods and the concentration-compactness principle, we demonstrate the existence and multiplicity of solutions.