Kirchhoff Type Problems Research Articles

Existence results of positive solutions for Kirchhoff type biharmonic equationvia bifurcation methods

This paper is concerned with the existence of positive solutions for the fourth order Kirchhoff type problem $$ \left\{\begin{array}{ll} \Delta^{2}u-(a+b\int_\Omega \nabla u ^2dx)\triangle u=\lambda f(u(x)),\ \ \text{in}\ \Omega,\\ u=\triangle u=0,\ \ \text{on}\ \partial\Omega,\\ \end{array} \right. $$ where $\Omega\subset \mathbb{R}^{N}$($N\geq 1$) is a bounded domain with smooth boundary $\partial \Omega$, $a>0, b\geq 0$ 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 elliptic equations 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 positive solution for all $\lambda>0$ and no positive solution for $\lambda

Existence and multiplicity of positive solutions for a class of Kirchhoff type problems with nonlinear boundary conditions

In this work, we peruse a class of Kirchhoff type problems with nonlinear boundary conditions on a bounded domain, and earn the uniqueness and existence of positive solutions for a class of those problems by the variational methods. Also, we earn the multiplicity of positive solutions of those problems by using the Nehari manifold.

Existence of sign-changing solutions for $p(x)$-Laplacian Kirchhoff type problem in $\mathbb{R}^N$

The $p(x)$-Laplacian Kirchhoff type equation involving the nonlocal term $b \int_{\mathbb{R}^N} (1/p(x)) \lvert\nabla u\rvert^{p(x)}dx$ is investigated. Based on the variational methods, deformation lemma and other technique of analysis, it is proved that the problem possesses one least energy sign-changing solution $u_b$ which has precisely two nodal domains. Moreover, the convergence property of $u_b$ as the parameter $b \searrow 0$ is also obtained.

On a Class of Critical p-Biharmonic Kirchhoff Type Problems with Indefinite Weights

Using the genus theory introduced by Krasnoselskii and a variant of the mountain pass theorem due to Rabinowitz [24], we study the existence of solutions for the following Kirchhoff type problem: $$\begin{aligned} {\left\{ \begin{array}{ll} M\left( \int _\Omega |\Delta u|^p\,\mathrm{d}x\right) \Delta \Big (|\Delta u|^{p{-}2}\Delta u\Big ) {=} \lambda |u|^{p^{**}{-}2}u{+}a(x)|u|^{p{-}2}u{+}f(x,u), \,\, x\in \Omega ,\\ u = \frac{\partial u}{\partial \nu } = 0,\,\,x\in \partial \Omega , \end{array}\right. } \end{aligned}$$ where $$\Omega $$ is a bounded domain in $$\mathbb {R}^N$$ ( $$N\ge 3$$ ) with $$C^2$$ boundary, $$1<p<\frac{N}{2}$$ , $$p^{**} = \frac{Np}{N-2p}$$ is the critical exponent, $$\Delta $$ is the Laplace operator and $$\frac{\partial u}{\partial \nu }$$ is the outer normal derivative, $$\lambda $$ is a positive parameter, $$M: [0,+\infty ) \rightarrow \mathbb {R}$$ and $$f: \overline{\Omega }\times \mathbb {R}\rightarrow \mathbb {R}$$ are continuous functions, and $$a\in L^\infty (\Omega )$$ is a weight function.

Existence and asymptotic behavior of positive solutions for Kirchhoff type problems with steep potential well

In this paper, we consider the following Kirchhoff type problem{−(a+b∫R3|∇u|2dx)Δu+λV(x)u=|u|p−2uinR3,u∈H1(R3), where a>0 is a constant, b and λ are positive parameters, and 2<p<6. Suppose that the nonnegative continuous potential V represents a potential well with the bottom V−1(0), the equation has been extensively studied in the case 4≤p<6. In contrast, no existence result of solutions is available for the case 2<p<4 due to the presence of the term (∫R3|∇u|2dx)Δu. By combining the truncation technique and the parameter-dependent compactness lemma, we prove the existence of positive solutions for b small and λ large in the case 2<p<4. Moreover, we also explore the decay rate of the positive solutions as |x|→∞ as well as their asymptotic behavior as b→0 and λ→∞.

Multiplicity results involving p-biharmonic Kirchhoff-type problems

This paper deals with the existence of multiple solutions for the following Kirchhoff type equations involving p-biharmonic operator:−M(∫Ω(|Δpu|2+|u|p)dx)(Δp2u−|u|p−2u)=λf(x)|u|q−2u+g(x)|u|m−2u,x∈Ω,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document} $$\\begin{aligned}& -M \\biggl( \\int_{\\varOmega} \\bigl( \\vert \\Delta_{p}u \\vert ^{2}+ \\vert u \\vert ^{p} \\bigr)\\,dx \\biggr) \\bigl( \\Delta _{p}^{2}u- \\vert u \\vert ^{p-2}u \\bigr) =\\lambda f(x) \\vert u \\vert ^{q-2}u+g(x) \\vert u \\vert ^{m-2}u,\\quad x\\in\\varOmega, \\end{aligned}$$ \\end{document} where Ω is a bounded domain in mathbb{R}^{N} (N>1), lambda >0, p, q, m>1, M is a continuous function, and the weight functions f and g are measurable. We obtain the existence results by combining the variational method with Nehari manifold and fibering maps.

Existence of three weak solutions for Kirchhoff-type problems with variable exponent and nonhomogeneous Neumann conditions

Existence of three weak solutions for Kirchhoff-type problems with variable exponent and nonhomogeneous Neumann conditions

On𝓅(x)-Kirchhoff-type equation involving𝓅(x)-biharmonic operator via genus theor

UDC 517.9 The paper deals with the existence and multiplicity of nontrivial weak solutions for the𝓅(x)-Kirchhoff-type problem,u=Δu=0on∂Ω.By using variational approach and Krasnoselskii’s genus theory, we prove the existence and multiplicity of solutions for the𝓅(x)-Kirchhoff-type equation.

Nodal solution for Kirchhoff-type problems with concave-convex nonlinearities

In this paper, we study the existence of nodal solutions for the Kirchhoff-type problem with concave-convex nonlinearities where Ω is a bounded domain with smooth boundary in , a . By constraining the energy functional of the above problem on a subset of the nodal Nehari set , we show that there exists a constant such that for any , the above problem has a nodal solution with positive energy.

Semi-classical solutions for Kirchhoff type problem with a critical frequency

AbstractIn the present paper, we consider the following Kirchhoff type problem$$ -\Big(\varepsilon^2+\varepsilon b \int_{\mathbb R^3} | \nabla v|^2\Big) \Delta v+V(x)v=|v|^{p-2}v \quad {\rm in}\ \mathbb{R}^3, $$whereb&gt; 0,p∈ (4, 6), the potential$V\in C(\mathbb R^3,\mathbb R)$andɛis a positive parameter. The existence and multiplicity of semi-classical state solutions are obtained by variational method for this problem with several classes of critical frequency potentials, i.e.,$\inf _{\mathbb R^N} V=0$. As to Kirchhoff type problem, little has been done for the critical frequency cases in the literature, especially the potential may vanish at infinity.

Infinitely many solutions for a new Kirchhoff-type equation with subcritical exponent

ABSTRACT In this article, we consider the following new Kirchhoff-type problem: where a and b are positive constants, is a bounded domain with boundary , with if , and if N = 1, 2. We show that the problem possesses infinitely many sign-changing solutions by using combination of invariant sets of descent flow and the Ljusternik–Schnirelman type minimax method. And an example for p = 2 is illustrated our results.

General fractional Sobolev space with variable exponent and applications to nonlocal problems

In this paper, we extend the fractional Sobolev spaces with variable exponents $$W^{s,p(x,y)}$$ to include the general fractional case $$W^{s,p(x,y)}_K$$ , where p is a variable exponent, $$s\in (0,1)$$ and K is a suitable kernel. We are concerned with some qualitative properties of the space $$W^{s,p(x,y)}_K$$ (completeness, reflexivity, separability, and density). Moreover, we prove a continuous and a compact embedding theorem of these spaces into variable exponent Lebesgue spaces. As applications, we discuss the existence of a nontrivial solution for a nonlocal p(x, .)-Kirchhoff type problem. Further, we establish the existence and uniqueness of a solution for a variational problem involving the integro-differential operator of elliptic type $${\mathcal {L}}^{p(x,.)}_K$$ .

Positive solutions for a Kirchhoff-type problem involving multiple competitive potentials and critical Sobolev exponent

In this paper, we study the existence and multiplicity of positive solutions for a Kirchhoff-type problem involving several potentials and critical nonlinearities on unbounded domain. The first main result is that the problem possesses a ground state solution concentrating around a concrete set relates to the potentials. The second one is that with the help of Nehari manifold and Ljusternik–Schnirelmann category, the relationship between the number of positive solutions and the category of the global minima set of a suitable ground energy function is established. Moreover, we present a sufficient condition for the nonexistence of ground state solution.

On the Kirchhoff problems involving critical Sobolev exponent

This paper deals with the following Kirchhoff type problem (0.1)−a+b∫RN|∇u|2Δu=λK(x)uq−1+u2∗−1,u>0,x∈RN,where N≥3, K∈L1(RN)∩L∞(RN), constants a,b>0, the parameter λ>0 and 2≤q<2∗≔2N∕(N−2). Using the finite-dimensional reduction approach, we prove that problem (0.1) admits at least a positive solution.

Existence of solution for a class of nonvariational Kirchhoff type problem via dynamical methods

In this paper, we use dynamical methods to establish the existence of nontrivial solution for the following class of Kirchhoff type problem (P)−1+b∫Ω|∇u|2dxΔu=f(u)+h(u,∇u),x∈Ω,u=0,x∈∂Ω, where Ω⊂RN(N=2,3) is a smooth bounded domain, b≥0, f:R→R and h:R×RN→[0,+∞) are C1-functions that satisfy some technical conditions.

Nonlocal Kirchhoff Problems with Singular Exponential Nonlinearity

In this paper, we first develop the fractional Trudinger–Moser inequality in singular case and then we use it to study the existence and multiplicity of solutions for a class of perturbed fractional Kirchhoff type problems with singular exponential nonlinearity. Under some suitable assumptions, the existence of two nontrivial and nonnegative solutions is obtained by using the mountain pass theorem and Ekeland’s variational principle as the nonlinear term satisfies critical or subcritical exponential growth conditions. Moreover, the existence of ground state solutions for the aforementioned problems without perturbation and without the Ambrosetti–Rabinowitz condition is investigated.

On a fractional Kirchhoff type problem with critical exponential growth nonlinearity

This paper is devoted to study the existence of nontrivial solutions to a fractional Kirchhoff type problem with the nonlinearity term having critical exponential growth, which is related to the fractional Trudinger–Moser type inequality.

Multiplicity of concentrating positive solutions for nonlinear Kirchhoff type problems with critical growth

ABSTRACT In this paper, we study the following semilinear Kirchhoff type equation: where is a parameter, a, b are positive constants, V and f are continuous functions. Under certain assumptions on V and f, we investigate the relation between the number of solutions and the topology of the set where V attains its local minimum for small ε. We also describe the concentration phenomena of solutions as . The proof is based on the method of Nehari manifold, penalization techniques and Ljusternik–Schnirelmann category theory.

Existence and Multiplicity of Positive Solutions for Kirchhoff-Type Equations with the Critical Sobolev Exponent

In this paper, we consider the following Kirchhoff-type problems involving critical exponent −a+b∫Ω∇u2dxΔu+Vxu=μu2∗−1+λgx,u, x∈Ωu&gt;0, x∈Ωu=0, x∈∂Ω. The existence and multiplicity of positive solutions for Kirchhoff-type equations with a nonlinearity in the critical growth are studied under some suitable assumptions on Vx and gx,u. By using the mountain pass theorem and Brézis–Lieb lemma, the existence and multiplicity of positive solutions are obtained.

Two weak solutions for some Kirchhoff-type problem with Neumann boundary condition

Abstract In this article we prove the existence of at least two weak solutions for a Kirchhoff-type problem by using the minimum principle, the mountain pass theorem and variational methods in Orlicz–Sobolev spaces.