Kirchhoff Type Problems Research Articles

On the Brezis–Nirenberg problem for a Kirchhoff type equation in high dimension

The present paper deals with a parametrized Kirchhoff type problem involving a critical nonlinearity in high dimension. Existence, non existence and multiplicity of solutions are obtained under the effect of a subcritical perturbation by combining variational properties with a careful analysis of the fiber maps of the energy functional associated to the problem. The particular case of a pure power perturbation is also addressed. Through the study of the Nehari manifolds we extend the general case to a wider range of the parameters.

Weak Nontrivial Solutions to Discrete Nonlinear Two-Point Boundary-Value Problems of Kirchhoff Type

"We prove the existence of at least one weak nontrivial solutions for a discrete nonlinear two-point boundary-value problems of Kirchhoff type. The main existence results are obtained by using the technique of geometric mountain pass and the Ekelands variational principle."

A New Approach to the Hyers–Ulam–Rassias Stability of Differential Equations

Based on the lower and upper solutions method, we propose a new approach to the Hyers–Ulam and Hyers–Ulam–Rassias stability of first-order ordinary differential equations $$u'=f(t,u)$$ , in the lack of Lipschitz continuity assumption. Apart from extending and improving the literature by dropping some assumptions, our result provides an estimate for the difference between the solutions of the exact and perturbed models better than from that one obtained by fixed point approach which is commonly used method in this topic. Some examples are also given to illustrate the improvement. Particularly, we examine our approach to the Hyers–Ulam stability problem of second-order elliptic differential equations $$-\Delta u =g(x,u)$$ with homogeneous Dirichlet boundary condition which arise in different applications such as population dynamics and population genetics. This investigation is not only of interest in its own right, but also it supports the usability of our approach to other types of boundary value problems such as p(x)-Laplacian Dirichlet problems, Kirchhoff type problems, fractional differential equations and etc.

Existence and Uniqueness of Weak Solutions to Variable-Order Fractional Laplacian Equations with Variable Exponents

In this paper, the variable-order fractional Laplacian equations with variable exponents and the Kirchhoff-type problem driven by p · -fractional Laplace with variable exponents were studied. By using variational method, the authors obtain the existence and uniqueness results.

Existence of positive solutions for a class of semipositone problems with Kirchhoff operator

In this work we show a result of existence of positive solution for the following nonlocal problem of Kirchhoff type \(-M\biggl(\int_{\Omega} |\nabla u|^{2} dx\biggl)\Delta u= f(u)-a\) in \(\Omega\), \(u=0\) on \(\partial\Omega,\) where \(\Omega \subset\mathbf{R}^{N}\) is a smooth bounded domain, \(M,f\) are continuous nonnegative functions and \(a>0\). By using mainly variational methods, we prove the existence of a solution for \(a\) small enough, under two different sets of hypotheses, which generalize the classical superlinear and sublinear problems.

Two nonzero weak solutions for a quasilinear Kirchhoff type problem

Two nonzero weak solutions for a quasilinear Kirchhoff type problem

Multiplicity and concentration of positive solutions to the fractional Kirchhoff type problems involving sign-changing weight functions

<p style='text-indent:20px;'>The aim of this paper is to study the multiplicity and concentration of positive solutions to the fractional Kirchhoff type problems involving sign-changing weight functions and concave-convex nonlinearities with subcritical or critical growth. Applying Nehari manifold, fibering maps and Ljusternik-Schnirelmann theory, we investigate a relationship between the number of positive solutions and the topology of the global maximum set of <inline-formula><tex-math id="M1">\begin{document}$ K $\end{document}</tex-math></inline-formula>.</p>

Two weak solutions for fully nonlinear Kirchhoff-type problem

In this article, the existence of solutions for fully nonlinear Kirchhoff-type problem -M (??(|?u|) + ?(|u|))dx) [div(a(|?u|)?u) + a(|u|)u] = ? ?ki=1 (tqi(x)-1- tri(x)-1)is proved via variational method. Finally, some new problems are introduced.

Existence results for a Kirchhoff-type problem with singularity

Existence results for a Kirchhoff-type problem with singularity

Sign-changing solutions for a class of fractional Kirchhoff-type problem with logarithmic nonlinearity

In this paper, we are interested the following fractional Kirchhoff-type problem with logarithmic nonlinearity $ \left\{ \begin{array} {ll} \left(a+b \iint_{\Omega^2} \frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}} dxdy\right)(-\Delta)^s u + V(x)u = Q(x) |u|^{p-2}u \ln u^2, & {\rm in } \ \Omega, \\ u = 0, & {\rm in } \ \mathbb{R}^N \setminus \Omega, \end{array} \right. $ where $\Omega \subset \mathbb{R}^N$ is a smooth bounded domain, $N > 2s$ ($0 < s < 1$), $(-\Delta)^s$ is the fractional Laplacian, $V, Q$ are continuous, $V, Q \ge 0$. $a, b > 0$ are constants, $4 < p < 2_s^* : = \frac{2N}{N-2s}$. By using constraint variational method, a quantitative deformation lemma and some analysis techniques, we obtain the existence of ground state sign-changing solutions for above problem.

On a fourth-order elliptic Kirchhoff type problem with critical Sobolev exponent

This work is concerned with a class of fourth-order elliptic Kirchhoff type problems involving the critical term. By means of the truncation and the concentration compact argument, for each positive integer k the existence of $k$ pairs nontrivial solutions is established.

Multiplicity of solutions for some singular quasilinear Schrödinger–Kirchhoff equations with critical exponents

ABSTRACT In this paper, we focus on the existence of multiplicity of solutions for the singular quasilnear Schrödinger–Kirchhoff type problem: where , a>0, , , 1<p<2, , with , , and we get the existence of infinitely many weak solutions by variational methods.

Existence and multiplicity results for Kirchhoff-type problem with sublinear nonlinearity

In this paper, we deal with the existence and multiplicity of solutions of Kirchhoff-type problem (0.1)−a+b∫RN|∇u|2dx△u+u=f(x,u),inRN,u∈H1(RN),where N≥3. Under more relaxed assumptions on f, we establish some existence criteria to guarantee that the above problem has at least one or infinitely many nontrival solutions by using the genus properties in critical point theory.

A class of diffusion problem of Kirchhoff type with viscoelastic term involving the fractional Laplacian

This work is concerned with a class of diffusion problem of Kirchhoff type with viscoelastic term and nonlinear interior source in the setting of the fractional Laplacian. Under suitable conditions we prove the existence of global solutions and the exponential decay of the energy.

Existence of solutions for a p-Laplacian Kirchhoff type problem with nonlinear term of superlinear and subcritical growth

"This paper is concerned by the study of the existence of nonnegative and nonpositive solutions for a nonlocal quasilinear Kirchhoff problem by using the Mountain Pass lemma technique."

Existence and multiplicity results for critical anisotropic Kirchhoff-type problems with nonlocal nonlinearities

In this paper, we are interested in existence and multiplicity of solutions to anisotropic elliptic equations of Kirchhoff-type given by where Ω is a smooth bounded domain of and are parameters, where is the critical exponent and . The function is only continuous with and can change its sign, is a function with subcritical growth and Under appropriate assumptions on f, applying an adequate truncation argument on M and the Concentration Compactness-Principle for the anisotropic operator [1], we obtain the existence of a nontrivial solution for by the Mountain Pass Theorem [2] and multiplicity of solutions by Krasnoselskii's genus and the symmetric Mountain Pass Lemma [3]. As a model case for M and f, we can consider or with and with and appropriate

On Critical Schrödinger–Kirchhoff-Type Problems Involving the Fractional p-Laplacian with Potential Vanishing at Infinity

The aim of this paper is to study the existence of solutions for critical Schrodinger–Kirchhoff-type problems involving a nonlocal integro-differential operator with potential vanishing at infinity. As a particular case, we consider the following fractional problem: $$\begin{aligned}&M\left( \iint _{{\mathbb {R}}^{2N}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}\mathrm{dxdy}+\int _{{\mathbb {R}}^N}V(x)|u(x)|^{p}\mathrm{{d}}x\right) ((-\Delta )_p^{s}u(x)+V(x)|u|^{p-2}u)\\&\quad =K(x)(\lambda f(x,u)+|u|^{p_s^{*}-2}u), \end{aligned}$$ where $$M:[0, \infty )\rightarrow [0, \infty )$$ is a continuous function, $$(-\Delta )_p^{s}$$ is the fractional p-Laplacian, $$0<s<1<p<\infty $$ with $$sp<N,$$ $$p_s^{*}=Np/(N-ps),$$ K, V are nonnegative continuous functions satisfying some conditions, and f is a continuous function on $${\mathbb {R}}^N\times {\mathbb {R}}$$ satisfying the Ambrosetti–Rabinowitz-type condition, $$\lambda >0$$ is a real parameter. Using the mountain pass theorem, we obtain the existence of the above problem in suitable space W. For this, we first study the properties of the embedding from W into $$L_{K}^{\alpha }({\mathbb {R}}^N), \alpha \in [p, p_s^{*}].$$ Then, we obtain the differentiability of energy functional with some suitable conditions on f. To the best of our knowledge, this is the first existence results for degenerate Kirchhoff-type problems involving the fractional p-Laplacian with potential vanishing at infinity. Finally, we fill some gaps of papers of do O et al. (Commun Contemp Math 18: 150063, 2016) and Li et al. (Mediterr J Math 14: 80, 2017).

Existence of ground state solutions for Kirchhoff type problem without the Ambrosetti–Rabinowitz condition

In this paper, we prove the existence of a positive ground state solution for Kirchhoff type problem with critical exponential growth without the Ambrosetti–Rabinowitz condition in dimension two.

On a Kirchhoff diffusion equation with integral condition

This paper is devoted to Kirchhoff-type parabolic problem with nonlocal integral condition. Our problem has many applications in modeling physical and biological phenomena. The first part of our paper concerns the local existence of the mild solution in Hilbert scales. Our results can be studied into two cases: homogeneous case and inhomogeneous case. In order to overcome difficulties, we applied Banach fixed point theorem and some new techniques on Sobolev spaces. The second part of the paper is to derive the ill-posedness of the mild solution in the sense of Hadamard.

Existence of Ground States for Kirchhoff-Type Problems with General Potentials

In this paper, we consider the following Kirchhoff-type problem: $$\begin{aligned} \left\{ \begin{array}{ll} -\left( a+b\int _{\mathbb {R}^3}|\nabla u|^2\mathrm {d}x\right) \triangle u+V(x)u=f(u), &{} x\in \mathbb {R}^3; \\ u\in H^1(\mathbb {R}^3), \end{array} \right. \end{aligned}$$ where $$a, b> 0$$ , $$V\in {\mathcal {C}}(\mathbb {R}^3, \mathbb {R})$$ and $$f\in {\mathcal {C}}(\mathbb {R}, \mathbb {R})$$ . Using variational method and some new analytical techniques, we show the existence of ground state solutions for the above problem. Assumptions imposed on the potential V and the nonlinearity f are general, and they are satisfied by several functions. Our results generalize and improve the ones obtained recently in [Li and Ye, J. Differential Equations (2014)], [Tang and Chen, Calc. Var. Partial Differential Equations (2017)], [Guo, J. Differential Equations (2015)] and some other related literature.