Elliptic Equations Of Kirchhoff Type Research Articles

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}.

Asymptotics of Solution Curves of Kirchhoff Type Elliptic Equations with Logarithmic Kirchhoff Function

We study the one-dimensional nonlocal elliptic equation of Kirchhoff type with logarithmic Kirchhoff function. We establish the precise asymptotic formulas for the solution $$u_\lambda (x)$$ as $$\lambda \rightarrow \infty $$ . Here, $$\lambda > 0$$ is the bifurcation parameter.

Global bifurcation curves of nonlocal elliptic equations with oscillatory nonlinear term

We study the one-dimensional nonlocal elliptic equation of Kirchhoff type with oscillatory nonlinear term. We establish the precise asymptotic formulas for the bifurcation curves λ ( α ) as α → ∞ and α → 0 , where α := ‖ u λ ‖ ∞ and u λ is the solution associated with λ . We show that the second term of λ ( α ) is oscillatory as α → ∞ .

Weigthed elliptic equation of Kirchhoff type with exponential non linear growthWeigthed elliptic equation of Kirchhoff type with exponential non linear growth

"This work is concerned with the existence of a positive ground state solution for the following non local weighted problem \begin{equation*} \displaystyle \left\{ \begin{array}{rclll} L_{(\sigma,V)}u &= & \displaystyle f(x,u)& \mbox{in} \ B \\ u &>&0 &\mbox{in }B\\ u&=&0 &\mbox{on } \partial B, \end{array} \right. \end{equation*} where $$L_{(\sigma,V)}u:=g(\int_{B}(\sigma(x)|\nabla u|^{N}+V(x)|u|^{N})dx)\big[-\textmd{div} (\sigma(x)|\nabla u|^{N-2} \nabla u)+V(x)|u|^{N-2}u\big],$$ B is the unit ball of $\mathbb{R}^{N}$, $ N>2$, $\sigma(x)=\Big(\log(\frac{e}{|x|})\Big)^{\beta(N-1)}$, $\beta \in[0,1)$ the singular logarithm weight , $V(x)$ is a positif continuous potential.The Kirchhoff function $g$ is positive and continuous on $(0,+\infty)$. The nonlinearities are critical or subcritical growth in view of Trudinger-Moser inequalities of exponential type. We prove the existence of a positive ground state solution by using Mountain Pass theorem . In the critical case, the Euler-Lagrange function loses compactness except for a certain level. We dodge this problem by using adapted test functions to identify this level of compactness."

Nodal solutions of fourth-order Kirchhoff equations with critical growth in R^N

We consider a class of fourth-order elliptic equations of Kirchhoff type with critical growth in \(R^N\). By using constrained minimization in the Nehari manifold, weestablish sufficient conditions for the existence of nodal (that is, sign-changing) solutions.
 For more information see https://ejde.math.txstate.edu/Volumes/2021/19/abstr.html

Sign-changing solutions for fourth-order elliptic equations of Kirchhoff type with critical exponent

Sign-changing solutions for fourth-order elliptic equations of Kirchhoff type with critical exponent

Multiple Solutions for Critical Fourth-Order Elliptic Equations of Kirchhoff type

In this paper, we study the existence and multiplicity of solutions for a class of equations involving a nonlocal term and the biharmonic operator with critical exponent. By new techniques, multiplicity results are established together with variational method. It is worth noting that the method of this paper is obviously different from the existing literature.

Multiplicity of solutions to the generalized extensible beam equations with critical growth

In this paper by variational methods, we study the multiplicity of solutions to the following fourth-order elliptic equations of Kirchhoff type with critical nonlinearity in RN: Δ2u−M∫RN|∇u|2dxΔu+V(x)u=k(x)|u|q−2u+λ|u|2∗∗−2u,x∈RN, where Δ2u=Δ(Δu) is the biharmonic operator, M:R+→R+ is a continuous function, λ>0 is a parameter, V:RN→R+ is a potential function, and k(x) is a nonnegative continuous real valued function satisfying some conditions. The compactness condition is proved by the Lions’ second Concentration-compactness principle and Concentration-compactness principle at infinity.

Infinitely Many High Energy Solutions for a Fourth-Order Equations of Kirchhoff Type in ℝN

In this paper we study the following fourth-order elliptic equations of Kirchhoff type $$\Delta^2u - (a+b\int_{\mathbb{R}^N} | \triangledown u|^2dx)\Delta u + V(x)u=f(x, u), \;\;x\in\mathbb{R}^N,$$ where Δ2 := Δ(Δ) is the biharmonic operator, a, b > 0 are constants, V ∈ C(ℝN, ℝ) and f ∈ C(ℝN × ℝ, ℝ). Under some appropriate assumptions on V(x) and f(x, u), new results on the existence of infinitely many high energy solutions for the above equation are obtained via Symmetric Mountain Pass Theorem.

Multiplicity of Radially Symmetric Small Energy Solutions for Quasilinear Elliptic Equations Involving Nonhomogeneous Operators

We investigate the multiplicity of radially symmetric solutions for the quasilinear elliptic equation of Kirchhoff type. This paper is devoted to the study of the L ∞ -bound of solutions to the problem above by applying De Giorgi’s iteration method and the localization method. Employing this, we provide the existence of multiple small energy radially symmetric solutions whose L ∞ -norms converge to zero. We utilize the modified functional method and the dual fountain theorem as the main tools.

Multiple solutions for a fourth-order elliptic equation of Kirchhoff type with variable exponent

In this paper, we consider a class of fourth-order elliptic equations of Kirchhoff type with variable exponent [Formula: see text] where [Formula: see text], [Formula: see text], is a smooth bounded domain, [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] is the operator of fourth-order called the [Formula: see text]-biharmonic operator, [Formula: see text] is the [Formula: see text]-Laplacian, [Formula: see text] is a log-Hölder continuous function and [Formula: see text] is a continuous function satisfying some certain conditions. A multiplicity result for the problem is obtained by using the mountain pass theorem and Ekeland’s variational principle provided [Formula: see text] is small enough.

On the existence of solutions for a class of fourth order elliptic equations of Kirchhoff type with variable exponent

In this paper, we consider a class of fourth order elliptic equations of Kirchhoff type with variable exponent$$\left\{\begin{array}{ll}\Delta^2_{p(x)}u-M\left(\int_\Omega\frac{1}{p(x)}|\nabla u|^{p(x)}\,dx\right)\Delta_{p(x)} u = \lambda f(x,u) \quad \text{ in }\Omega,\\u=\Delta u = 0 \quad \text{ on } \partial\Omega, \end{array}\right.$$where $\Omega \subset \R^N$, $N \geq 3$, is a smooth bounded domain, $M(t)=a+bt^\kappa$, $a, \kappa>0$, $b \geq 0$, $\lambda$ is a positive parameter, $\Delta_{p(x)}^2u=\Delta (|\Delta u|^{p(x)-2} \Delta u)$ is the operator of fourth order called the $p(x)$-biharmonic operator, $\Delta_{p(x)}u = \operatorname{div} \left(|\nabla u|^{p(x)-2}\nabla u\right)$ is the $p(x)$-Laplacian, $p:\overline\Omega \to \R$ is a log-H\"{o}lder continuous function and $f: \overline\Omega\times \R\to \R$ is a continuous function satisfying some certain conditions. Using Ekeland's variational principle combined with variational techniques, an existence result is established in an appropriate function space.

Existence of Nonnegative Solutions for Fourth Order Elliptic Equations of Kirchhoff Type with General Subcritical Growth

This paper is dedicated to investigating the following fourth-order elliptic equation with Kirchhoff-type \[ \begin{cases} \displaystyle \Delta^{2} u - \left( a + b \int_{\mathbb{R}^{N}} |\nabla u|^{2} \, dx \right) \Delta u + cu = f(u) &\textrm{in $\mathbb{R}^{N}$}, \\ u \in H^{2}(\mathbb{R}^{N}), \end{cases} \] where $a \gt 0$, $b \geq 0$ and $c \gt 0$ are constants. By using cut-off functional and monotonicity tricks, we prove that the above problem has a positive solution. Our result cover the case where the nonlinearity satisfies asymptotically linear and superlinear condition at the infinity, which extend the results of related literatures.

Nontrivial solutions for a fourth-order elliptic equation of Kirchhoff type via Galerkin method

In this paper we study the existence of nontrivial solutions of the nonlocal elliptic problem $$\begin{aligned} \left\{ \begin{array}{lcr} \Delta ^2 u-(a+b\int _\Omega |\nabla u|^2\mathrm{d}x)\Delta u+u=(\int _\Omega g(x,u)\mathrm{d}x)^\gamma f(x,u),\;in\;\Omega ,\\ u=\Delta u=0,\;\;on\;\;\partial \Omega \end{array}\right. \end{aligned}$$ via Galerkin method, where $$\Omega \subset \mathbb {R}^N (N\ge 5)$$ is a smooth bounded domain.

On Fourth-Order Elliptic Equations of Kirchhoff Type with Dependence on the Gradient and the Laplacian

We consider a nonlocal fourth-order elliptic equation of Kirchhoff type with dependence on the gradient and Laplacian Δ2u-a+b∫Ω∇u2dxΔu=fx,u,∇u,Δu, in Ω, u=0, Δu=0, on ∂Ω, where a, b are positive constants. We will show that there exists b⁎>0 such that the problem has a nontrivial solution for 0<b<b⁎ through an iterative method based on the mountain pass lemma and truncation method developed by De Figueiredo et al., 2004.

Existence of Solution for a Singular Elliptic Equation of Kirchhoff Type

We study both the existence and uniqueness of nonnegative solution to a singular elliptic problem of Kirchhoff type, whose model is: $$\begin{aligned} {\left\{ \begin{array}{ll} -B\left( \dfrac{1}{2}\displaystyle \int _\Omega |\nabla u|^2\mathrm {d}x\right) \Delta u=\dfrac{h(x)}{u^\gamma }, &{}\quad x\in \Omega ,\\ u>0, &{}\quad x\in \Omega ,\\ u=0, &{}\quad x\in \partial \Omega , \end{array}\right. } \end{aligned}$$ where $$\Omega \subset \mathbb {R}^n(n\ge 1)$$ is a smooth bounded domain, $$\gamma >1$$ , $$h\in L^1(\Omega )$$ is positive (i.e., $$h(x)>0$$ a.e. in $$\Omega $$ ), $$B : \mathbb {R}^+\rightarrow \mathbb {R}^+$$ is a $$C^1$$ -continuous function with positive lower bound. A necessary and sufficient condition will be given for the existence of weak solution of the general nonlocal singular elliptic with strong singularity. In addition, we prove that the solution is unique under some suitable conditions.

Nontrivial solutions of nonlocal fourth order elliptic equation of Kirchhoff type in [formula omitted

This paper is devoted to a class of important and general nonlocal fourth order elliptic problemΔ2u−(1+λ∫R3|∇u|2dx)Δu+V(x)u=f(x,u) in R3, where Δ2=Δ(Δ) is the bi-harmonic operator, λ≥0 is a constant. We focus on the case that f(x,u) involves a combination of convex and concave terms and the potential V(x) is allowed to be sign-changing. By new techniques, multiplicity results of two different type of solutions are established. Our results improves and generalizes that obtained in the literature.

Existence of positive solution for nonlocal singular fourth order Kirchhoff equation with Hardy potential

This paper is concerned with the existence of positive solution to a class of singular fourth order elliptic equation of Kirchhoff type $$\begin{aligned} \triangle ^2 u-\lambda M(\Vert \nabla u\Vert ^2)\triangle u-\frac{\mu }{\vert x\vert ^4}u=\frac{h(x)}{u^\gamma }+k(x)u^\alpha , \end{aligned}$$ under Navier boundary conditions, \(u=\triangle u=0\). Here \(\varOmega \subset {\mathbf {R}}^N\), \(N\ge 1\) is a bounded \(C^4\)-domain, \(0\in \varOmega \), h(x) and k(x) are positive continuous functions, \(\gamma \in (0,1)\), \(\alpha \in (0,1)\) and \(M:{\mathbf {R}}^+\rightarrow {\mathbf {R}}^+\) is a continuous function. By using Galerkin method and sharp angle lemma, we will show that this problem has a positive solution for \(\lambda > \frac{\mu }{\mu ^*m_0}\) and \(0 0\) and h, k are Lipschitz functions, we show that this problem has a positive smooth solution. If \(h,k\in C^{2,\,\theta _0}(\overline{\varOmega })\) for some \(\theta _0\in (0,1)\), then this problem has a positive classical solution.

Existence and multiplicity of solutions for fourth-order elliptic equations of Kirchhoff type with critical growth in ℝN

In this paper, we deal with the existence and multiplicity of solutions for fourth-order elliptic equations of Kirchhoff type with critical nonlinearity: −ε4Δ2u+ε4a+b∫RN∇u2dxΔu+V(x)u=u2**−2u+h(x,u), (t, x) ∈ ℝ × ℝN. By using Lions’ second concentration-compactness principle and concentration-compactness principle at infinity to prove that (PS) condition holds locally and by variational method, we prove that it has at least one solution and for any m ∈ ℕ, it has at least m pairs of solutions.

Multiplicity of Solutions for Fourth-Order Elliptic Equations of Kirchhoff Type with Critical Exponent

In this paper, we deal with fourth-order elliptic equations of Kirchhoff type with critical exponent in bounded domains, the new results about existence, and multiplicity of solutions are obtained by using the concentration-compactness principle and variational method.