Kirchhoff Type Problems Research Articles

Virtual element method for a nonlocal elliptic problem of Kirchhoff type on polygonal meshes

We consider the discretization of the nonlocal elliptic problem of Kirchhoff type using the virtual element method (VEM) over polygonal meshes. The nonlocal diffusion coefficient is approximated by using the L2 projection operator, which is directly computable from the degrees of freedom. However, the presence of the nonlocal term reduces the sparsity of the Jacobian matrix, which would increase the computational burden. To avoid this and to retain the sparsity of the Jacobian, the nonlinear system is replaced by an equivalent system inspired by the work on the FEM by Gudi (2012). The numerical results show that the proposed methodology yields optimal convergence rate in the L2 norm. Theoretical estimates are derived for H1 norms and are verified by solving a numerical example.

A Class of Critical Magnetic Fractional Kirchhoff Problems

In this paper, we deal with the existence and asymptotic behavior of solutions for a fractional Kirchhoff type problem involving the electromagnetic fields and critical nonlinearity by using the classical critical point theorem. Meanwhile, an example is given to illustrate the application of the main result.

Infinitely many solutions for a class of $p(x)$-Kirchhoff type problems with critical exponents

Infinitely many solutions for a class of $p(x)$-Kirchhoff type problems with critical exponents

Ground States of Nonlinear Schrödinger-Kirchhoff Type Equation

In this paper, we considered with the following Schr?dinger-Kirchhoff type problem: -(a+b ∫RN|∇u|2dx)Δu + V(x)u = f(x,u) in RN. We put forward general assumptions on the nonlinearity f with the subcritical growth and we find a ground state solution being a minimizer of the energy functional associated with a Nehari-Pankov manifold by using a linking theorem.

Existence and Multiplicity of Solutions to Discrete Inclusions with the p(k)-Laplacian Problem of Kirchhoff Type

Existence and Multiplicity of Solutions to Discrete Inclusions with the p(k)-Laplacian Problem of Kirchhoff Type

Instability of solutions to Kirchhoff type problems in low dimension

Instability of solutions to Kirchhoff type problems in low dimension

Existence and uniqueness of positive solutions for Kirchhoff type beam equations

This paper is concerned with the existence and uniqueness of positive solution for the fourth order Kirchhoff type problem $$\left\{\begin{array}{ll} u''''(x)-(a+b\int_0^1(u'(x))^2dx)u''(x)=\lambda f(u(x)), x\in(0,1), u(0)=u(1)=u''(0)=u''(1)=0, \end{array} \right. $$ where $a>0, b\geq 0$ are constants, $\lambda\in \mathbb{R}$ is a parameter. For the case $f(u)\equiv u$, we use an argument based on the linear eigenvalue problems of fourth order equations and their properties to show that there exists a unique positive solution for all $\lambda>\lambda_{1,a}$, here $\lambda_{1,a}$ is the first eigenvalue of the above problem with $b=0$; For the case $f$ is sublinear, we prove that there exists a unique positive solution for all $\lambda>0$ and no positive solution for $\lambda<0$ by using bifurcation method.

Sign-changing solutions for a class of &lt;em&gt;p&lt;/em&gt;-Laplacian Kirchhoff-type problem with logarithmic nonlinearity

In this paper, we study the existence of ground state sign-changing solutions for following $p$-Laplacian Kirchhoff-type problem with logarithmic nonlinearity $ \begin{equation*} \left\{ \renewcommand{\arraystretch}{1.25} \begin{array}{ll} -(a+ b\int _{\Omega}|\nabla u|^{p}dx)\Delta_p u = |u|^{q-2}u\ln u^2, ~x\in\Omega \\ u = 0, ~ x\in \partial\Omega, \end{array} \right. \end{equation*} $ where $\Omega\subset \mathbb{R}^{N}$ is a smooth bounded domain, $a, b \gt 0$ are constant, $4\leq 2 p \lt q \lt p^*$ and $N \gt p$. By using constraint variational method, topological degree theory and the quantitative deformation lemma, we prove the existence of ground state sign-changing solutions with precisely two nodal domains.

Existence and concentration of ground state solutions for a class of Kirchhoff-type problems

This paper is concerned with the following singularly perturbed Kirchhoff-type problem −ε2a+εb∫R3|∇u|2dx△u+V(x)u=f(u),x∈R3;u∈H1(R3),where ε>0 is a small parameter, a,b>0 are two constants, V∈C(R3,R), and f∈C(R,R) is of super-linear growth at infinity and satisfies neither the usual Ambrosetti–Rabinowitz type condition nor monotonicity condition on f(u)∕u3. By using some new techniques and subtle analyses, we prove that there exists a constant ε0>0 determined by V and f such that for ε∈(0,ε0], the above problem has a ground state solution concentrating around global minimum of V in the semi-classical limit. Our results are available to the case that f(u)∼|u|s−2u for s∈(2,6), and extend the existing results concerning the case that f(u)∼|u|s−2u for s∈[4,6).

Bifurcation results for a Kirchhoff type problem involving sign-changing weight functions

In this paper we establish some existence results for both compact and unbounded connected components of the positive solutions set of a nonlocal problem of the Kirchhoff type. The methods to show our main results are the local and global bifurcation techniques, a priori bounds for elliptic equation, and the properties of the principal eigenvalues. To obtain the a priori bounds for Kirchhoff type elliptic equation, some new techniques are introduced.

Berestycki-Lions conditions on ground state solutions for Kirchhoff-type problems with variable potentials

Based on the strategy introduced by Chen and Tang [Adv. Nonlinear Anal. 9, 496–515 (2020)] and some new tricks, we prove that the nonlinear problem of Kirchhoff-type −a+b∫R3|∇u|2dx△u+V(x)u=f(u), x∈R3 in H1(R3) admits two classes of ground state solutions under the general “Berestycki-Lions assumptions” on the nonlinearity f, which are almost necessary conditions, as well as some weak assumptions on the potential V. Moreover, we also give a simple minimax characterization of the ground state energy. Our results improve and extend recent theorems in several directions.

Least energy nodal solution for Kirchhoff type problem with an asymptotically 4-linear nonlinearity

In the present paper, we consider the following Kirchhoff type problem −(a+b∫R3|∇u|2dx)Δu+V(x)u=f(u)inR3,where a,b>0 and the potential V has double potential well. With an asymptotically 4-linear nonlinearity f, we prove that the Kirchhoff type problem has one least energy nodal solution by variational methods.

Infinitely many radial solutions for a super-cubic Kirchhoff type problem in a ball

We prove the existence of infinitely many radial solutions to a Kirchhoff type problem in a ball with a super-cubic nonlinearity. Our methods rely on bifurcation analysis and energy estimates.

On a critical Kirchhoff-type problem

In the present paper we study a Kirchhoff type problem involving the critical Sobolev exponent. We give sufficient conditions for the sequentially weakly lower semicontinuity and the Palais–Smale property of the energy functional associated to the problem.

Existence of positive solutions for the nonlinear Kirchhoff type equations in [formula omitted

In this paper, we consider the following nonlinear problem of Kirchhoff type:{−(a+λ∫Rn|∇u|2dx)Δu+V(x)u=|u|p−1u,x∈Rn,u∈H1(Rn),u>0, where n≥3, a, λ are positive constants, 1<p<max⁡(3,n+2n−2) and V(x) is a positive continuous potential satisfying V0≤V(x)≤lim inf|x|→∞V(x)=V∞. Using variational methods and a cutoff technique, we prove the existence of positive solution to the above equation for all λ>0 small.

Existence and multiplicity of solutions for fractional p(x,.)-Kirchhoff-type problems in ℝ N

The purpose of this paper is to investigate the existence and multiplicity of solutions for the following class of fractional -Kirchhoff-type problems in We first establish the compact embedding of the space into . Using this embedding result and some critical-points theorems, we obtain the existence and multiplicity results.

Sign-changing and nontrivial solutions for a class of Kirchhoff-type problems

In this paper, we study the existence of sign-changing (nodal) and nontrivial solutions for the nonlinear Kirchhoff-type equation{−(a+b∫Ω|∇u|2dx)Δu=αu+βu3inΩ,u=0on∂Ω, where α,β∈R are two real parameters. With the help of nodal Nehari set, we first provide a description of a two-dimensional set in the (α,β) plane, which corresponds to the nonexistence and existence of sign-changing solutions for the above Kirchhoff-type equation. And then, we establish the existence result of nontrivial solutions via the minimax methods.

Existence of solutions for quasilinear Kirchhoff type problems with critical nonlinearity in $\mathbb {R}^N$

We study the existence of solutions for a class of Kirchhoff type problems with critical growth in $\mathbb {R}^N$:$$-\varepsilon ^2\biggl (a+b\int _{\mathbb {R}^N}|\nabla u|^2\,dx\biggr )\Delta u + V(x)u -\varepsilon ^2a\Delta (u^2)u = |u|^{22^\ast -2}u + h(x,u),$$ $(t, x) \in \mathbb {R} \times \mathbb {R}^N$. By using a change of variables, the quasilinear equations are reduced to a semilinear one, whose associated functionals are well defined in the usual Sobolev space and satisfy the geometric conditions of the mountain pass theorem for suitable assumptions. We prove that it has at least one solution and for any $m \in \mathbb {N}$, it has at least $m$ pairs of solutions. The proofs are based on the variational methods and concentration-compactness principle.

Three solutions for a Kirchhoff-type problem involving nonlocal fractional $p$-Laplacian

‎‎Using the three critical points theorem‎, ‎we obtain the existence of three weak solutions for a Kirchhoff-type problem involving the nonlocal fractional $p$-Laplacian operator in a fractional Sobolev space‎, ‎with homogeneous Dirichlet boundary conditions‎.

Global Existence and Blow-Up for a Parabolic Problem of Kirchhoff Type with Logarithmic Nonlinearity

In this paper, we study the following parabolic problem of Kirchhoff type with logarithmic nonlinearity: $$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle {u_t} +{M([u]^2_s){\mathcal {L}}_Ku}={|u|^{p-2}u\log |u|},\ \ \ &{}\hbox { in } \Omega \times (0,+\infty ),\\ \displaystyle u(x,t)=0,&{}\hbox { in }({\mathbb {R}}^N\setminus \Omega )\times (0,+\infty ),\\ \displaystyle u(x,0)=u_0(x),&{}\hbox { in }\Omega , \end{array}\right. \end{aligned}$$ where $$[u]_s$$ is the Gagliardo seminorm of u, $$\Omega \subset {\mathbb {R}}^N$$ is a bounded domain with Lipschitz boundary, $$0<s<1$$ , $${\mathcal {L}}_K$$ is a nonlocal integro-differential operator defined in (1.2), which generalizes the fractional Laplace operator $$(-\Delta )^s$$ , $$u_0$$ is the initial function, and $$M:[0,+\infty )\rightarrow [0,+\infty )$$ is continuous. Let $$J(u_0)$$ be the initial energy (see (2.1) for the definition of J), $$d>0$$ be the mountain-pass level given in (2.4), and $${\widetilde{M}}\in (0,d]$$ be the constant defined in (2.6). Firstly, we get the conditions on global existence and finite time blow-up for $$J(u_0)\le d$$ . Then we study the lower and upper bounds of blow-up time to blow-up solutions under some appropriate conditions. Secondly, for $$J(u_0)\le {\widetilde{M}}$$ , the growth rate of the solution is got. Moreover, we give some blow-up conditions independent of d and study the upper bound of the blow-up time. Thirdly, the behavior of the energy functional as $$t\rightarrow T$$ is also discussed, where T is the blow-up time. In addition, for $$J(u_0)\le d$$ , we give some equivalent conditions for the solutions blowing up in finite time or existing globally. Finally, we consider the existence of ground state solutions and the asymptotical behavior of the general global solution.