Kirchhoff-type System Research Articles

Existence and classification of positive solutions for coupled purely critical Kirchhoff system

We study the nonlinear coupled Kirchhoff system with purely Sobolev critical exponent. By using appropriate transformation, we get one equivalent system involving a critical Schrödinger system and an algebraic system. Through solving the critical Schrödinger system with a corresponding algebraic system, under suitable conditions we obtain the existence and classification of positive ground states for the Kirchhoff system in dimensions 3 and 4. Furthermore, for the degenerate case, we give a complete classification of positive ground states for the Kirchhoff system in any dimension. To the best of our knowledge, this paper is the first to give classification results for the ground states of Kirchhoff systems. The results in this paper partially extend and complement the main results established by Lü and Peng [Existence and asymptotic behavior of vector solutions for coupled nonlinear Kirchhoff-type system, J. Differ. Equ. 263 (2017) 8947–8978] considering the linearly coupled Kirchhoff system with subcritical exponent and some partial results established by Chen and Zou [Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent, Arch. Ration. Mech. Anal. 205 (2012) 515–551; Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent: higher dimensional case, Calc. Var. Partial Differ. Equ. 52 (2015) 423–467], where the authors considered the coupled purely critical Schrödinger system.

Elliptic Kirchhoff-type system with two convections terms and under Dirichlet boundary conditions

This work discusses the existence of weak solutions for a system of Kirchhoff-type involving variable exponent (?1(m), ?2(m))-Laplacian operators and under the Dirichlet boundary conditions. Under appropriate hypotheses on the nonlinear terms and the Kirchhoff functions, the existence of weak solutions is obtained on the spaces W1,?1(m) 0 (D) ? W1,?2(m) 0 (D). The proof of the main result is based on a topological degree argument for a class of demicontinuous operators of (S+)-type.

An Existence Result for a Class of Kirchhoff Type Systems via Dynamical Methods

An Existence Result for a Class of Kirchhoff Type Systems via Dynamical Methods

Existence of multi-bump solutions for a nonlinear Kirchhoff-type system

In this paper, we use the Lyapunov–Schmidt reduction method to obtain the existence of multi-bump solutions for a nonlinear Kirchhoff-type system with the parameter ɛ. As a result, when ɛ → 0, the system has more and more multi-bump positive solutions.

Normalized solutions for nonlinear fractional Kirchhoff type systems

Normalized solutions for nonlinear fractional Kirchhoff type systems

A Class of Fourth-Order Symmetrical Kirchhoff Type Systems

This paper deals with the existence and multiplicity of solutions for a perturbed nonlocal fourth-order class of p(·)&q(·)-Kirchhoff elliptic systems under Navier boundary conditions. By using the variational method and Ricceri’s critical point theorem, we can find a proper conditions to ensure that the perturbed fourth-order of (p(x),q(x))-Kirchhoff systems has at least three weak solutions. We have extended and improved some recent results. The complexity of the combination of variable exponent theory and fourth-order Kirchhoff systems makes the results of this work novel and new contribution. Finally, a very concrete example is given to illustrate the applications of our results.

Degenerate Fractional Kirchhoff-Type System with Magnetic Fields and Upper Critical Growth

This paper deals with the following degenerate fractional Kirchhoff-type system with magnetic fields and critical growth: $$\begin{aligned} \left\{ \begin{array}{lll} -\mathfrak {M}(\Vert u\Vert _{s,A}^2)[(-\Delta )^s_Au+u] = G_u(|x|,|u|^2,|v|^2) \\ \quad +\left( \mathcal {I}_\mu *|u|^{p^*}\right) |u|^{p^*-2}u \ &{}\text{ in }\,\,\mathbb {R}^N,\\ \mathfrak {M}(\Vert v\Vert _{s,A})[(-\Delta )^s_Av+v] = G_v(|x|,|u|^2,|v|^2) \\ \quad +\left( \mathcal {I}_\mu *|v|^{p^*}\right) |v|^{p^*-2}v \ &{}\text{ in }\,\,\mathbb {R}^N, \end{array}\right. \end{aligned}$$where $$\begin{aligned}\Vert u\Vert _{s,A}=\left( \iint _{\mathbb {R}^{2N}}\frac{|u(x)-e^{i(x-y)\cdot A(\frac{x+y}{2})}u(y)|^2}{|x-y|^{N+2s}}{\text {d}}x {\text {d}}y+\int _{\mathbb {R}^N}|u|^2{\text {d}}x\right) ^{1/2},\end{aligned}$$and \((-\Delta )_{A}^s\) and A are called magnetic operator and magnetic potential, respectively, \(\mathfrak {M}:\mathbb {R}^{+}_{0}\rightarrow \mathbb {R}^{+}_0\) is a continuous Kirchhoff function, \(\mathcal {I}_\mu (x) = |x|^{N-\mu }\) with \(0<\mu <N\), \(C^1\)-function G satisfies some suitable conditions, and \(p^* =\frac{N+\mu }{N-2s}\). We prove the multiplicity results for this problem using the limit index theory. The novelty of our work is the appearance of convolution terms and critical nonlinearities. To overcome the difficulty caused by degenerate Kirchhoff function and critical nonlinearity, we introduce several analytical tools and the fractional version concentration-compactness principles which are useful tools for proving the compactness condition.

On initial inverse problem for nonlinear couple heat with Kirchhoff type

The main objective of the paper is to study the final model for the Kirchhoff-type parabolic system. Such type problems have many applications in physical and biological phenomena. Under some smoothness of the final Cauchy data, we prove that the problem has a unique mild solution. The main tool is Banach’s fixed point theorem. We also consider the non-well-posed problem in the Hadamard sense. Finally, we apply truncation method to regularize our problem. The paper is motivated by the work of Tuan, Nam, and Nhat [Comput. Math. Appl. 77(1):15–33, 2019].

Infinitely many solutions for nonlocal elliptic systems in Orlicz–Sobolev spaces

Abstract Recently, the existence of at least two weak solutions for a Kirchhoff–type problem has been studied in [M. Makvand Chaharlang and A. Razani, Two weak solutions for some Kirchhoff-type problem with Neumann boundary condition, Georgian Math. J. 28 2021, 3, 429–438]. Here, the existence of infinitely many solutions for nonlocal Kirchhoff-type systems including Dirichlet boundary conditions in Orlicz–Sobolev spaces is studied by using variational methods and critical point theory.

Existence of Solutions for a Class of Fractional Kirchhoff-type Systems in $\mathbb{R}^N$ with Non-standard Growth

This paper is concerned with the existence and multiplicity of nontrivial solutions for a class of Kirchhoff-type systems in $\mathbb{R}^N$ involving the fractional pseudo-differential operators defined as the generalizations of the $p(x)$-Laplace operator. Our main tools come from a direct variational methods, the Mountain Pass Theorem, the symmetric Mountain Pass Theorem and the Fountain Theorem in critical point theory. The obtained results of this note significantly contribute to the study of Kirchhoff-type systems in the sense that our situation covers not only differential operators of fractional order but also nonhomogeneous differential operators in Sobolev spaces with variable exponent.

Normalized ground state solutions for Kirchhoff type systems

We consider the existence of ground state solutions for nonlinear Kirchhoff type systems in the whole space RN (2 ≤ N ≤ 4) with prescribed normalization. Two cases are studied: one is L2-supercritical and the other is mixed. In the first case, assuming that the coupling coefficient is big enough, we prove the existence of a ground state solution via minimax methods. In the second case, assuming that the coupling coefficient is sufficiently small, we show the existence of a local minimizer, which is, of course, also a ground state solution.

On a Class of Elliptic (p, q)-Kirchhoff Type Systems with Multiple Parameters

This article concerns the existence of positive solutions for elliptic (p, q)-Kirchhoff type systems with multiple parameters. Our approach is based on the method of sub and super-solutions. The concepts of sub- and super-solution were introduced by Nagumo (Proc Phys-Math Soc Jpn19:861–866, 1937) in 1937 who proved, using also the shooting method, the existence of at least one solution for a class of nonlinear Sturm-Liouville problems. In fact, the premises of the sub- and super-solution method can be traced back to Picard. He applied, in the early 1880s, the method of successive approximations to argue the existence of solutions for nonlinear elliptic equations that are suitable perturbations of uniquely solvable linear problems. This is the starting point of the use of sub- and super-solutions in connection with monotone methods. Picard’s techniques were applied later by Poincare (J Math Pures Appl 4:137–230, 1898) in connection with problems arising in astrophysics. We refer to Rădulescu (Qualitative analysis ofnonlinear elliptic partial differential equations: monotonicity, analytic, and variational methods, contemporary mathematics and its applications, Hindawi Publishing Corporation, New York, 2008).

Existence of solutions for Kirchhoff type systems involving Q-Laplacian operator in Heisenberg group

Let Hn be the Heisenberg group, Q=2n+2 be the homogeneous dimension of Hn, we study the existence of solutions for the following Q-Laplacian elliptic system{−K(∫Ω|∇Hnu|Qdξ)ΔQu=λGu(ξ,u,v)ρ(ξ)℘inΩ;−K(∫Ω|∇Hnv|Qdξ)ΔQv=λGv(ξ,u,v)ρ(ξ)℘inΩ;u=0,v=0,on∂Ω, where Ω is an open, smooth and bounded subset of Heisenberg group Hn, K is a Kirchhoff type function, 0≤℘<Q and λ is a positive parameter, and nonlinear terms Gu,Gv have critical exponential growth behave like exp⁡(β|s|QQ−1) as |s|→+∞ with some β>0.

Existence of solutions for a critical (p 1(x), … , pn(x))-Kirchhoff-type potential systems

ABSTRACT In this paper, by using the concentration–compactness principle of Lions for variable exponent spaces and the Mountain Pass Theorem, we obtain the existence of nontrivial solutions for a class of -Kirchhoff-type potential systems type with critical exponent.

Existence of multiple positive solutions for a truncated Kirchhoff-type system involving weight functions and concave\u2013convex nonlinearities

We consider the combined effect of concave–convex nonlinearities on the number of solutions for an indefinite truncated Kirchhoff-type system involving the weight functions. When alpha+ beta<4, since the concave-convex nonlinearities do not satisfy the mountain pass geometry, it is difficult to obtain a bounded Palais–Smale sequence by the usual mountain pass theorem. To overcome the problem, we properly introduce a method of Nehari manifold and then establish the existence of multiple positive solutions when the pair of the parameters is under a certain range.

Existence of Solutions for Kirchhoff-Type Fractional Dirichlet Problem with p-Laplacian

In this paper, we investigate the existence of solutions for a class of p-Laplacian fractional order Kirchhoff-type system with Riemann–Liouville fractional derivatives and a parameter λ . By mountain pass theorem, we obtain that system has at least one non-trivial weak solution u λ under some local conditions for each given large parameter λ . We get a concrete lower bound of the parameter λ , and then obtain two estimates of weak solutions u λ . We also obtain that u λ → 0 if λ tends to ∞. Finally, we present an example as an application of our results.

Existence of least energy nodal solution for Kirchhoff-type system with Hartree-type nonlinearity

This paper deals with following Kirchhoff-type system with critical growth $ \begin{cases} -(a+ b\int _{\mathbb{R}^3}|\nabla u|^{2}dx)\Delta u+ V(x)u+\phi|u|^{p-2}u = |u|^{4}u+\mu f(u), ~ x\in\mathbb{R}^3, (-\Delta)^{\alpha/2}\phi = l|u|^p, ~ x\in \mathbb{R}^3, \end{cases} $ where $a, \mu \gt 0$, $b, l\geq0$, $\alpha\in(0, 3)$, $p\in[2, 3)$ and $\phi|u|^{p-2}u$ is a Hartree-type nonlinearity. By the minimization argument on the nodal Nehari manifold and the quantitative deformation lemma, we prove that the above system has a least energy nodal solution. Our result improve and generalize some interesting results which were obtained in subcritical case.

Existence Results for a Class of Kirchhoff-Type Systems with Combined Nonlinear Effects

We study the existence of positive solutions for a nonlinear system$$ {\displaystyle \begin{array}{l}-{M}_1\left(\underset{\varOmega }{\int }{\left|{\nabla}_u\right|}^p dx\right)\operatorname{div}\left({\left|x\right|}^{- ap}{\left|{\nabla}_u\right|}^{p-2}{\nabla}_u\right)=\lambda {\left|x\right|}^{-\left(a+1\right)p+{c}_1}f\left(u,\upsilon \right),\kern1em x\in \varOmega, \\ {}-{M}_2\left(\underset{\varOmega }{\int }{\left|{\nabla}_{\upsilon}\right|}^q dx\right)\operatorname{div}\left({\left|x\right|}^{- bq}{\left|{\nabla}_{\upsilon}\right|}^{q-2}{\nabla}_{\upsilon}\right)=\lambda {\left|x\right|}^{-\left(b+1\right)q+{c}_2}g\left(u,\upsilon \right),\kern1em x\in \varOmega, \\ {}u=\upsilon =0,\kern1em x\in \vartheta \varOmega, \end{array}} $$ where Ω is a bounded smooth domain in ℝN with $$ 0\in \varOmega, 1\kern0.33em <\kern0.33em p,q\kern0.33em <N,0\le a<\frac{N-p}{p},0\le b<\frac{N-q}{q}, $$ and c1, c2, and λ are positive parameters. Here, M1, M2, f, and g satisfy certain conditions. We use the method of sub- and supersolutions to establish our results.

Kirchhoff-type system with linear weak damping and logarithmic nonlinearities

For the nonlinear Kirchhoff-type wave system with logarithmic nonlinearities and weak dissipation the global well-posedness of initial boundary value problem is analyzed. Focusing on the interplay between Kirchhoff terms and logarithmic sources, we investigate the Kirchhoff system controlled by logarithmic forces thus amplifying the difficulties in blow up analysis which is the primary scenario of interest. By employing potential well method and concavity method, we obtain several results related to the sufficient conditions posed on subcritical initial energy and critical initial energy, which is used to classify initial data for global existence and finite time blow up. Finally, via careful analysis involving the unstable invariant set under supercritical initial energy, we are able to show an affirmative result that the solution blows up in finite time when initial data satisfy some suitable assumptions.

Existence of positive solutions for a class of Kirchhoff type systems involving critical exponents

In this paper, we consider a class of Kirchhoff type systems involving critical exponents in bounded domains. Under appropriate conditions on the nonlinearities, we prove the existence and asymptotic behavior of positive solutions for the problem by using truncation argument combined with the mountain pass theorem and a variant of concentration compactness principle related to critical elliptic systems in [19].