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We investigate the existence of solutions of the Kirchhoff elliptic equations with nonlinearity in $R^{N}$ . Using the ideas developed in Orlicz spaces and the technique of variation principle, we prove that there are at least three solutions in Orlicz-Sobolev spaces. [ABSTRACT FROM AUTHOR]

The initial boundary value problem of a class of Kirchhoff equations with strong damping and nonlinear dissipation is considered. By modifying Vitillaro's argument, we prove a blow-up result for solutions with positive and negative initial energy respectively. [ABSTRACT FROM AUTHOR]

We consider a weak viscoelastic Kirchhoff plate model with time-varying delay in the boundary. By using a suitable energy and Lyapunov function, we obtain a decay rate for the energy, which depends on the behavior of g and α. [ABSTRACT FROM AUTHOR]

We investigate the existence of multiple solutions for perturbed nonlocal fourth-order equations of Kirchhoff type under Navier boundary conditions. We give some new criteria for guaranteeing that the perturbed fourth-order equations of Kirchhoff type have at least three weak solutions by using a variational method and some critical point theorems due to Ricceri. We extend and improve some recent results. Finally, by presenting two examples, we ensure the applicability of our results. [ABSTRACT FROM AUTHOR]

In this paper we consider the existence and general energy decay rate of global solution to the mixed problem for the Kirchhoff-type equation with memory boundary and acoustic boundary conditions. In order to prove the existence of solutions, we employ the Galerkin method and compactness arguments. Besides, we establish an explicit and general decay rate result using the perturbed modified energy method and some properties of the convex functions. Our result is obtained without imposing any restrictive assumptions on the behavior of the relaxation function at infinity. These general decay estimates extend and improve some earlier results, i.e., exponential or polynomial decay rates. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following p-Kirchhoff equation: with Dirichlet boundary conditions, where Ω is a bounded domain in $\mathbb{R}^{N}$ . Under proper assumptions on M and f, we obtain three existence theorems of infinitely many solutions for problem (P) by the fountain theorem. Moreover, for a special nonlinearity $f(x,u)=\lambda |u|^{q-2}u+|u|^{r-2}u$ ( $1< q< p< r< p^{*}$ ), we prove that problem (P) has at least two nonnegative solutions via the Nehari manifold method and a sequence of solutions with negative energy by the dual fountain theorem. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of p-Kirchhoff type problems with a singularity in a bounded domain in $R^{N}$ . By using the variational method, the existence and multiplicity of positive solutions are obtained. [ABSTRACT FROM AUTHOR]

In this paper, we develop a mixed Legendre-Galerkin spectral method to approximate the buckling problem of simply supported Kirchhoff plates subjected to general plane stress tensor. By the spectral theory of compact operators, the rigorous error estimates for the approximate eigenvalues and eigenfunctions are provided. Finally, we present some numerical experiments which support our theoretical results. [ABSTRACT FROM AUTHOR]

In the paper, we are concerned with the system of Kirchhoff-Schrödinger-Poisson system under certain assumptions on $V_{1}$ , $V_{2}$ , K and f. We are interested in the existence of least energy sign-changing solutions to the system on $\mathbb{R}^{N}$ . Because two kinds of nonlocal terms $\phi_{u}$ and $\int_{\mathbb{R}^{N}}|\nabla u|^{2}$ are involved in the system, the methods are different from the Kirchhoff or the Schrödinger-Poisson system. The two nonlocal terms $\int_{\mathbb{R}^{N}}|\nabla u|^{2}$ and $\phi_{u}$ make that the functional $J(u^{+}+u^{-})\neq J(u^{+})+J(u^{-})$ . Moreover, the nonlocal term $\phi_{u}$ does not have the convergence property because of the assumption $V_{2}$ . In addition, the convergence of these two nonlocal terms are different. In the present paper, we unify the increasing property conditions on sign-changing solution in previous papers. We construct a new homotopy operator and then weaken the assumption that f is $C^{1}$ to that of f being only continuous. We prove that the system has a sign-changing solution via a constraint variational method combining with Brouwer's degree theory. [ABSTRACT FROM AUTHOR]

This paper is concerned with the following Kirchhoff-type problems with a nonsmooth potential: -(a + b ∫Ω |∇u|² dx)Δu ∈ ∂j(x, u) for a.a. x Ω, u = 0 on ∂Ω Using the nonsmooth mountain pass theorem, the nonsmooth local linking theorem, and the nonsmooth fountain theorem, we establish the existence and multiplicity of solutions for the problem. All this is based on the nonsmooth critical point. Some recent results in the literature are generalized and improved. [ABSTRACT FROM AUTHOR]

By using variational methods, we study the multiplicity of solutions for Kirchhoff type problems -(a + b ∫ ω | ∇u|²) Δ|u = f (x,u), in Ω u = 0, on ∂ Ω. Existence results of two nontrivial solutions and infinite many solutions are obtained. [ABSTRACT FROM AUTHOR]

In this paper, we study a fractional Kirchhoff type equation with Hardy–Littlewood–Sobolev critical exponent. By using variational methods, we obtain the existence of mountain-pass type solution and negative energy solutions. Also, we prove some further properties of solutions. [ABSTRACT FROM AUTHOR]

This paper is concerned with the following nonlinear fractional Kirchhoff equation: (a+b∫R3|(−Δ)s2u|2dx)(−Δ)su+V(x)u=f(u),x∈R3, where s∈(34,1), a>0, b≥0, (−Δ)s denotes the fractional Laplacian operator. Based on the methods of mountain pass theorem and invariant sets of descending flow, the existence of a positive solution, a negative solution, and multiple sign-changing solutions is obtained. [ABSTRACT FROM AUTHOR]

In this paper, we study the following superlinear p-Kirchhoff-type equation: {M(∫R2N|u(x)−u(y)|p|x−y|N+psdxdy)(−△)psu(x)−λ|u|p−2u=g(x,u)in Ω,u=0in RN∖Ω, Under suitable assumptions on g(x,u) without the (AR) condition, the existence of infinitely many solutions for the Kirchhoff equation of a fractional p-Laplacian is obtained by using the fountain theorem. Our conclusions generalize and extend some existing results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the fractional p-Kirchhoff -type system: {M(∫R2N|u(x)−u(y)|p|x−y|N+psdxdy)(−Δ)psu=μg(x)|u|β−2u+aa+bh(x)|u|a−2u|v|b,in Ω,M(∫R2N|v(x)−v(y)|p|x−y|N+psdxdy)(−Δ)psv=σf(x)|v|β−2v+ba+bh(x)|v|b−2v|u|a,in Ω,u=v=0,in RN∖Ω, where Ω⊂RN is a smooth bounded domain, (−Δ)ps is the fractional p-Laplacian operator with 0 and ps. a>1, b>1 satisfy 2. 1<β, ps∗=NpN−ps is the fractional critical exponent. μ, σ are two real parameters. M(t)=k+λtτ, k>0, λ, τ≥0, τ=0 if and only if λ=0. The weight functions g, f, h change sign in Ω and satisfy suitable conditions. By using the Nehari manifold method, it is proved that the system has at least two solutions provided that 2 and (μ,σ) belongs to a certain subset of R2. Also, by using the mountain pass theorem, we prove that there exist λ1≥λ0 such that the system admits at least a nontrivial solution for λ∈(0,λ0) and no nontrivial solution for λ>λ1 under the assumptions μ=σ=0 and p. [ABSTRACT FROM AUTHOR]