Kirchhoff Type Problems Research Articles

Sign-Changing Solutions for Kirchhoff-Type Problems with Variable Exponent

This paper is devoted to study a class of Kirchhoff-type problems with variable exponent. By means of the perturbation technique, the method of invariant sets for the descending flow and necessary estimates and the existence of infinitely many sign-changing solutions to this problem are obtained.

Implicit Finite Element Scheme with a Penalty for a Nonlocal Parabolic Obstacle Problem of Kirchhoff Type

Implicit Finite Element Scheme with a Penalty for a Nonlocal Parabolic Obstacle Problem of Kirchhoff Type

Multiplicity of positive solutions for a Kirchhoff type problem without asymptotic conditions

In this paper, we are concerned with the multiplicity of positive solutions for the following Kirchhoff type problem \[ \begin{cases} -\bigg({\varepsilon}^2a+{\varepsilon}b\int_{\mathbb{R}^3} |\n u|^2dx\bigg)\Delta u+u=Q(x)|u|^{p-2}u, & x\in\mathbb{R}^3,\\ u\in H^1\big(\mathbb{R}^3\big), \quad u> 0, & x\in\mathbb{R}^3, \end{cases} \] where $\varepsilon> 0$ is a small parameter, $a,b> 0$ are constants, $4< p< 6$, $Q$ is a nonnegative continuous potential and does not satisfy any asymptotic condition. Combining Nehari manifold and concentration compactness principle, we study how the shape of the graph of $Q(x)$ affects the number of positive solutions.

Three solutions for discrete anisotropic Kirchhoff-type problems

Abstract In this article, using critical point theory and variational methods, we investigate the existence of at least three solutions for a class of double eigenvalue discrete anisotropic Kirchhoff-type problems. An example is presented to demonstrate the applicability of our main theoretical findings.

On double phase Kirchhoff problems with singular nonlinearity

AbstractIn this paper, we study multiplicity results for double phase problems of Kirchhoff type with right-hand sides that include a parametric singular term and a nonlinear term of subcritical growth. Under very general assumptions on the data, we prove the existence of at least two weak solutions that have different energy sign. Our treatment is based on the fibering method in form of the Nehari manifold. We point out that we cover both the nondegenerate as well as the degenerate Kirchhoff case in our setting.

Existence of Weak Solution for $p(x)$-Kirchhoff Type Problem Involving the $p(x)$-Laplacian-like Operator by Topological Degree

Existence of Weak Solution for $p(x)$-Kirchhoff Type Problem Involving the $p(x)$-Laplacian-like Operator by Topological Degree

Nonlocal Kirchhoff-type problem involving variable exponent and logarithmic nonlinearity on compact Riemannian manifolds

Nonlocal Kirchhoff-type problem involving variable exponent and logarithmic nonlinearity on compact Riemannian manifolds

On a class of Kirchhoff type problems with singular exponential nonlinearity

We study the following singular Kirchhoff type problem 
 \[\left( P\right) \left\{
 \begin{array} [c]{c}
 -m\left({\displaystyle\int\limits_{\Omega}}\left\vert \nabla u\right\vert ^{2}dx\right) \Delta u=h\left( u\right)
 \frac{e^{\alpha u^{2}}}{\left\vert x\right\vert ^{\beta}}\text{ \ \ \ in} \Omega,\\
 u=0 \text{on}\; \partial\Omega
 \end{array} \right.
 \]
 where $\Omega\subset\mathbb{R}^{2}$ is a bounded domain with smooth boundary and $0\in\Omega,$ $\beta\in\left[ 0,2\right)$, $\alpha>0$ and $m$ is a continuous function
 on $\mathbb{R}^{+}.$ Here, $h$ is a suitable preturbation of $e^{\alpha u^{2}}$ as $u\rightarrow\infty.$ In this paper, we prove the existence of solutions of
 $(P)$. Our tools are Trudinger-Moser inequality with a singular weight and the mountain pass theorem

Variable exponent q(m)-Kirchhoff-type problems with nonlocal terms and logarithmic nonlinearity on compact Riemannian manifolds

Variable exponent q(m)-Kirchhoff-type problems with nonlocal terms and logarithmic nonlinearity on compact Riemannian manifolds

Existence of radial sign-changing solutions for fractional Kirchhoff-type problems in R3

Existence of radial sign-changing solutions for fractional Kirchhoff-type problems in R3

The third solution for a Kirchhoff-type problem with a critical exponent

The aim of this paper is to analyze the multiplicity of solutions for a nonhomogeneous Kirchhoff-type problem on R4 with a critical exponent. The first two positive solutions are deduced by using the variational method, and the third solution is found with the help of Brezis-Lieb's lemma and Mazur's lemma. Moreover, we obtain a new compactness condition and improve on the existing results in the literature.

Existence of nonnegative nontrivial solutions for Kirchhoff type problems with variable exponent

This paper is devoted to study a class of Kirchhoff type problems with variable exponent. By means of perturbation technique, variational methods and a priori estimation, the existence of nonnegative nontrivial solutions to this problem is obtained.

Existence of positive solutions for Kirchhoff-type problem in exterior domains

AbstractWe consider the following Kirchhoff-type problem in an unbounded exterior domain $\Omega\subset\mathbb{R}^{3}$: (*) \begin{align} \left\{ \begin{array}{ll} -\left(a+b\displaystyle{\int}_{\Omega}|\nabla u|^{2}\,{\rm d}x\right)\triangle u+\lambda u=f(u), & x\in\Omega,\\ \\ u=0, & x\in\partial \Omega,\\ \end{array}\right. \end{align}where a > 0, $b\geq0$, and λ > 0 are constants, $\partial\Omega\neq\emptyset$, $\mathbb{R}^{3}\backslash\Omega$ is bounded, $u\in H_{0}^{1}(\Omega)$, and $f\in C^1(\mathbb{R},\mathbb{R})$ is subcritical and superlinear near infinity. Under some mild conditions, we prove that if \begin{equation*}-\Delta u+\lambda u=f(u), \qquad x\in \mathbb R^3 \end{equation*}has only finite number of positive solutions in $H^1(\mathbb R^3)$ and the diameter of the hole $\mathbb R^3\setminus \Omega$ is small enough, then the problem (*) admits a positive solution. Same conclusion holds true if Ω is fixed and λ > 0 is small. To our best knowledge, there is no similar result published in the literature concerning the existence of positive solutions to the above Kirchhoff equation in exterior domains.

Kirchhoff-Type Problems Involving Logarithmic Nonlinearity with Variable Exponent and Convection Term

Kirchhoff-Type Problems Involving Logarithmic Nonlinearity with Variable Exponent and Convection Term

Elliptic anisotropic Kirchhoff-type problems with singular term

Elliptic anisotropic Kirchhoff-type problems with singular term

Multiple solutions for nonlinear boundary value problems of Kirchhoff type on a double phase setting

This paper deals with some classes of Kirchhoff type problems on a double phase setting and with nonlinear boundary conditions. Under general assumptions, we provide multiplicity results for such problems in the case when the perturbations exhibit a suitable behavior in the origin and at infinity, or when they do not necessarily satisfy the Ambrosetti–Rabinowitz condition. To this aim, we combine variational methods, truncation arguments and topological tools.

Existence and multiplicity of solutions for fractional $ p(x) $-Kirchhoff-type problems

<abstract><p>In this paper, we deal with the existence and multiplicity of solutions for fractional $ p(x) $-Kirchhoff-type problems as follows:</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \left\{ \begin{array}{l}M\Big(\int_Q\frac{1}{p(x, y)}\frac{| v(x)-v(y)|^{p(x, y)}}{| x-y|^{d+sp(x, y)}}dxdy\Big)(-\Delta_{p(x)})^s v(x)\ \, \, \, \, \, \, \,\\ = \lambda| v(x)|^{r(x)-2}v(x), \;\;\;\;\;\;\, \, \, \, \, \, \, \, \, \;\;\;\, \, \;\;\;\, \, \, \, \, \, \, \, \, \;\;\;\;\;\;\;\;\;\, \, \, \, \, \, \, \, \, \, \, \;\; \;\;\;\, \, \, \, \, \, \, \, \, \text{in}\;\;\Omega, \\ v = 0, \;\;\;\, \, \, \, \, \, \, \, \, \;\;\;\, \, \, \, \, \, \, \, \, \;\;\;\, \, \, \, \, \, \, \, \, \;\;\;\, \, \, \, \, \, \, \, \, \;\;\;\, \, \, \, \, \, \, \, \;\;\;\, \, \, \, \, \, \, \, \, \;\;\;\;\;\, \, \, \, \, \, \, \, \, \, \;\;\;\, \, \, \, \, \, \, \, \, \, \, \, \, \, \text{in}\;\mathbb{R}^d\backslash\Omega, \end{array}\right. $\end{document} </tex-math></disp-formula></p> <p>where $ (-\triangle_{p(x)})^s $ is the fractional $ p(x) $-Laplacian. Different from the previous ones which have recently appeared, we weaken the condition of $ M $ and obtain the existence and multiplicity of solutions via the symmetric mountain pass theorem and the theory of the fractional Sobolev space with variable exponents.</p></abstract>

Multiplicity of solutions for a singular Kirchhoff-type problem

This paper deals with some Kirchhoff-type problems driven by a non-local integrodifferential operator of singular elliptic type with combined nonlinearities which generalizes the fractional Laplacian operator. Our main result is to give and prove the existence of weak solutions for such problems with homogeneous Dirichlet boundary conditions. The proof is based on a variational method, precisely, we use the Nehari manifold method and the analysis of the fibering maps.

Multiplicity of solutions to non-local problems of Kirchhoff type involving Hardy potential

<abstract><p>The aim of this paper is to establish the existence of a sequence of infinitely many small energy solutions to nonlocal problems of Kirchhoff type involving Hardy potential. To this end, we used the Dual Fountain Theorem as a key tool. In particular, we describe this multiplicity result on a class of the Kirchhoff coefficient and the nonlinear term which differ from previous related works. To the best of our belief, the present paper is the first attempt to obtain the multiplicity result for nonlocal problems of Kirchhoff type involving Hardy potential by utilizing the Dual Fountain Theorem.</p></abstract>

A ground state solution for a nonhomogeneous elliptic Kirchhoff type problem involving critical growth and Hardy term

A ground state solution for a nonhomogeneous elliptic Kirchhoff type problem involving critical growth and Hardy term