Kirchhoff Type Problems Research Articles

Nodal solutions for a weighted (N,p)-Kirchhoff-type problem with double exponential growth non-linearity

This paper investigates the existence of sign-changing solutions for a logarithmic weighted Kirchhoff [Formula: see text]-Laplacian problem in the unit ball [Formula: see text] of [Formula: see text] where [Formula: see text]. The non-linearity of the equation is assumed to have double exponential growth, which is motivated by Trudinger–Moser-type inequalities. To prove the existence of these solutions, we employ techniques such as constrained minimization within the Nehari set, a quantitative deformation lemma and results from degree theory.

Existence results for p-Kirchhoff type problems on closed manifolds

This paper is to establish the existence, uniqueness, and compactness of positive solutions for the p-Kirchhoff type problems on a closed Riemannian manifold (M, g) of dimension n ≥ 3. Both subcritical and critical cases are considered under certain conditions by variational methods. Our results generalize several previous works from the special case of p = 2 to the general case of 1 < p < n.

A new variational characterization of the Fučik spectrum for the Kirchhoff-type problem and applications

We focus on the equation −∫Ω|∇u|2Δu=α(u+)3+β(u−)3 in Ω with the boundary condition u = 0 on ∂Ω, where Ω is a smooth bounded domain in RN for N = 1, 2, 3, α,β∈R, u+ = max{u, 0} and u− = min{u, 0}. Denote the Fučik spectrum by Σ, which is defined as a set comprising those (α,β)∈R2 such that the above equation has a nontrivial solution. The main goal of this study is to provide a new variational characterization for a class of curves in Σ. As an application, we establish the existence results for nonresonance and resonance problems related to these curves.

Kirchhoff problems with logarithmic double phase operator: Existence and multiplicity results

In this paper, we focus on Kirchhoff type problems driven by a logarithmic double phase operator with variable exponents. Under very general assumptions on the nonlinearity and using variational tools, like the mountain pass theorem, we establish the existence of at least one nontrivial weak solution for the problem under consideration. Then, under different hypotheses on the reaction term, we are also able to derive a multiplicity result of solutions for our problem. We stress that in order to produce such a multiplicity result a key role is played by a variant of the symmetric mountain pass theorem.

On a Class of p(z)-Biharmonic Kirchhoff Type Problems with Indefinite Weight and No-Flow Boundary Condition

On a Class of p(z)-Biharmonic Kirchhoff Type Problems with Indefinite Weight and No-Flow Boundary Condition

Existence of solutions for a hyperbolic Kirchhoff-type problem with a volume constraint

In this study, we investigate the existence of weak solutions for a hyperbolic Kirchhoff-type problem constrained by volume. Our research is motivated by the complex nature of such problems, which involve non-local terms arising from the Kirchhoff term and the volume constraint. These non-local terms introduce significant challenges, complicating the application of traditional analytical methods and necessitating the development of innovative approaches for resolution. To address these challenges, we leverage the hyperbolic discrete Morse flow, a powerful framework that allows us to adapt traditional techniques to the unique characteristics of our problem. This framework is particularly suited for handling the intricacies of non-local terms and offers a structured way to examine the interplay between these terms and the volume constraint. Our approach includes a rigorous mathematical analysis combined with the development of novel techniques specifically designed to manage the complex interactions within the system. Through this comprehensive methodology, we aim to establish the necessary conditions under which weak solutions can exist for the given hyperbolic Kirchhoff-type problem. The findings from this study provide new insights into the behavior of such systems, potentially contributing to advancements in the broader field of partial differential equations and applied mathematics.

The sign-changing solutions for a class of Kirchhoff-type problems with critical Sobolev exponents in bounded domains

The sign-changing solutions for a class of Kirchhoff-type problems with critical Sobolev exponents in bounded domains

A study of a Kirchhoff-type p(x)-triharmonic problem involving two parameters

Our most significant aim is to guarantee multiplicity results for a Kirchhoff-type problem involving two parameters and p(x)-triharmonic operator subject to the Navier boundary condition. More precisely, two critical point theorems are exploited to attain main theorems.

A variational approach to a constrained hyperbolic Kirchhoff type problem with free boundary

A variational approach to a constrained hyperbolic Kirchhoff type problem with free boundary

Positive solutions of Kirchhoff type problems with critical growth on exterior domains

Positive solutions of Kirchhoff type problems with critical growth on exterior domains

A p(x)-Kirchhoff Type Problem Involving the p(x)-Laplacian-Like Operators With Dirichlet Boundary Condition

This paper deals with a class of p(x)-Kirchhoff type problems involving the p(x)-Laplacian-like operators, arising from the capillarity phenomena, depending on two real parameters with Dirichlet boundary conditions. Using a topological degree for a class of demicontinuous operators of generalized (S+), we prove the existence of weak solutions of this problem. Our results extend and generalize several corresponding results from the existing literature. Keywords: p(x)-Kirchhoff type problems, p(x)-Laplacian-like operators, weak solutions, variable exponent Sobolev spaces.

Three weak solutions for a class of fourth order p(x)-Kirchhoff type problem with Leray-Lions operators

In this work, we study the multiplicity of a weak solution for a fourth order p(x)-Kirchhoff type problem involving the Leray-Lions type operators with no flux boundary condition. By using variational approach and critical point theory, we determine an open interval of parameters for which our problem admits at least three distinct weak solutions.

Asymptotics for a parabolic problem of Kirchhoff type with singular critical exponential nonlinearity

AbstractThe main objective of this paper is to characterize stable sets based on the asymptotic behavior of solutions as goes to infinity for the following class of parabolic Kirchhoff equations: where is a bounded domain with a Lipschitz boundary, , , , , , and is the fractional ‐Laplacian operator, .

Positive solutions for the fractional Kirchhoff type problem in exterior domains

Positive solutions for the fractional Kirchhoff type problem in exterior domains

On a class of fractional p( x, y)−Kirchhoff type problems with indefinite weight

This paper is concerned with a class of fractional \(p(x,y)-\)Kirchhoff type problems with Dirichlet boundary data along with indefinite weight of the following form \begin{equation*} \left\lbrace\begin{array}{ll} M\left(\int_{Q}\frac{1}{p(x,y)}\frac{|u(x)-u(y)|^{p(x,y)}}{|x-y|^{N+sp(x,y)}}\,dx\,dy\right)\\ (-\triangle_{p(x)})^s+|u(x)|^{q(x)-2}u(x) & \\ =\lambda V(x)|u(x)|^{r(x)-2}u(x)& \text{in }\Omega,\\ u=0, & \text{in }\mathbb{R}^N\Omega. \end{array}\right. \end{equation*} By means of direct variational approach and Ekeland’s variational principle, we investigate the existence of nontrivial weak solutions for the above problem in case of the competition between the growth rates of functions \(p\) and \(r\) involved in above problem, this fact is essential in describing the set of eigenvalues of this problem.

Multiple Positive Solutions for Kirchhoff-Type Problems Involving Supercritical and Critical Terms

Multiple Positive Solutions for Kirchhoff-Type Problems Involving Supercritical and Critical Terms

Necessary and sufficient conditions for ground state solutions to planar Kirchhoff-type equations

In this paper, we are concerned with the ground states of the following planar Kirchhoff-type problem: \[ -\left(1+b\displaystyle\int_{\mathbb{R}^2}|\nabla u|^2\,{\rm d}x\right)\Delta u+\omega u=|u|^{p-2}u, \quad x\in\mathbb{R}^2. \] where $b,\, \omega >0$ are constants, $p>2$ . Based on variational methods, regularity theory and Schwarz symmetrization, the equivalence of ground state solutions for the above problem with the minimizers for some minimization problems is obtained. In particular, a new scale technique, together with Lagrange multipliers, is delicately employed to overcome some intrinsic difficulties.

Multiplicity and concentration of positive solutions to the double phase Kirchhoff type problems with critical growth

The aim of this paper is to study the multiplicity and concentration of positive solutions to the $(p,q)$ Kirchhoff-type problems involving a positive potential and a continuous nonlinearity with critical growth at infinity. Applying penalization techniques, truncation methods and the Lusternik-Schnirelmann theory, we investigate a relationship between the number of positive solutions and the topology of the set where the potential $V$ attains its minimum values.

On a critical fourth order Leray–Lions $$p(\cdot )$$-Kirchhoff type problem with no-flux boundary condition

On a critical fourth order Leray–Lions $$p(\cdot )$$-Kirchhoff type problem with no-flux boundary condition

Existence and multiplicity of solutions for p(x)-Kirchhoff type problems with nonhomogeneous Neumann conditions

Existence and multiplicity of solutions for p(x)-Kirchhoff type problems with nonhomogeneous Neumann conditions