• Journal of the Korean Mathematical Society
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• v.42 no.1
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• pp.37-52
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• 2005

We consider the existence of solutions of viscoelastic degenerate problem of Kirchhoff type with nonlinear boundary damping and memory term. Moreover, we consider the uniform decay of the energy for the problem.

• Kyungpook Mathematical Journal
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• v.58 no.3
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• pp.533-546
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• 2018

The object of this work is to study the existence of solutions for a class of p(x)-Kirchhoff type problem under no-flux boundary conditions with dependence on the gradient. We establish our results by using the degree theory for operators of ($S_+$) type in the framework of variable exponent Sobolev spaces.

• Journal of applied mathematics & informatics
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• v.32 no.1_2
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• pp.113-128
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• 2014

Consider a class of p(x)-Kirchhoff type equations of the form $$\left\{-M\left({\int}_{\Omega}\;\frac{1}{p(x)}{\mid}{\nabla}u{\mid}^{p(x)}\;dx\right)\;div\;({\mid}{\nabla}u{\mid}^{p(x)-2}{\nabla}u)={\lambda}V(x){\mid}u{\mid}^{q(x)-2}u\;in\;{\Omega},\\u=0\;on\;{\partial}{\Omega},$$ where p(x), $q(x){\in}C({\bar{\Omega}})$ with 1 < $p^-\;:=inf_{\Omega}\;p(x){\leq}p^+\;:=sup_{\Omega}p(x)$ < N, $M:{\mathbb{R}}^+{\rightarrow}{\mathbb{R}}^+$ is a continuous function that may be degenerate at zero, ${\lambda}$ is a positive parameter. Using variational method, we obtain some existence and multiplicity results for such problem in two cases when the weight function V (x) may change sign or not.

• Bulletin of the Korean Mathematical Society
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• v.52 no.2
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• pp.661-677
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• 2015

In this paper, we consider the following Kirchhoff-type Schr$\ddot{o}$dinger system $$\{-\(a_1+b_1{\int}_{\mathbb{R^3}}{\mid}{\nabla}u{\mid}^2dx\){\Delta}u+{\gamma}V(x)u=\frac{2{\alpha}}{{\alpha}+{\beta}}{\mid}u{\mid}^{\alpha-2}u{\mid}v{\mid}^{\beta}\;in\;\mathbb{R}^3,\\-\(a_2+b_2{\int}_{\mathbb{R^3}}{\mid}{\nabla}v{\mid}^2dx\){\Delta}v+{\gamma}W(x)v=\frac{2{\beta}}{{\alpha}+{\beta}}{\mid}u{\mid}^{\alpha}{\mid}v{\mid}^{\beta-2}v\;in\;\mathbb{R}^3,\\u,v{\in}H^1(\mathbb{R}^3),$$ where $a_i$ and $b_i$ are positive constants for i = 1, 2, ${\gamma}$ > 0 is a parameter, V (x) and W(x) are nonnegative continuous potential functions. By applying the Nehari manifold method and the concentration-compactness principle, we obtain the existence and concentration of ground state solutions when the parameter ${\gamma}$ is sufficiently large.

• Journal of the Korean Mathematical Society
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• v.53 no.1
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• pp.201-215
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• 2016

In this paper, we consider the following $Schr{\ddot{o}}dinger$-Kirchhoff-type equations $\[a+b\({\int}_{{\mathbb{R}}^N}({\mid}{\nabla}u{\mid}^2+V(x){\mid}u{\mid}^2)dx\)\][-{\Delta}u+V(x)u]=f(x,u)$, in ${\mathbb{R}}^N$. Under certain assumptions on V and f, some new criteria on the existence and multiplicity of nontrivial solutions are established by the Morse theory with local linking and the genus properties in critical point theory. Some results from the previously literature are significantly extended and complemented.

• East Asian mathematical journal
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• v.28 no.3
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• pp.339-345
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• 2012

In this paper, we study uniform exponential stabilization of the vibrations of the Kirchho type wave equation with acoustic boundary in a bounded domain in $R^n$. To stabilize the system, we incorporate separately, the passive viscous damping in the model as like Gannesh C. Gorain [1]. Energy decay rate is obtained by the exponential stability of solutions by using multiplier technique.

• Kyungpook Mathematical Journal
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• v.61 no.1
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• pp.1-10
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• 2021

This paper is concerned the finite time blow-up of solution for higher-order nonlinear Kirchhoff-type equation with a degenerate term and a source term. By an appropriate Lyapunov inequality, we prove the finite time blow-up of solution for equation (1.1) as a suitable conditions and the initial data satisfying ||Dmu0|| > B-(p+2)/(p-2q), E(0) < E1.

• Bulletin of the Korean Mathematical Society
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• v.43 no.2
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• pp.245-252
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• 2006

In this paper we study the uniform decay estimates of the energy for the nonlinear wave equation of Kirchhoff type $$y'(t)-M({\mid}{\nabla}y(t){\mid}^2){\triangle}y(t)\;+\;{\delta}y'(t)=f(t)$$ with the damping constant ${\delta} > 0$ in a bounded domain ${\Omega}\;{\subset}\;\mathbb{R}^n$.

• Nonlinear Functional Analysis and Applications
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• v.29 no.1
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• pp.35-45
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• 2024

In this paper, we study the existence and multiplicity of nontrivial solutions for a p-Kirchhoff equation involving critical Sobolev-Hardy exponent by using variational methods and we need to estimate the energy levels.

• Journal of the Korean Mathematical Society
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• v.58 no.5
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• pp.1181-1209
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• 2021

In the present paper, we are concerned with the existence of ground state sign-changing solutions for the following Schrödinger-Poisson-Kirchhoff system $$\;\{\begin{array}{lll}-(1+b{\normalsize\displaystyle\smashmargin{2}{\int\nolimits_{{\mathbb{R}}^3}}}{\mid}{\nabla}u{\mid}^2dx){\Delta}u+V(x)u+k(x){\phi}u={\lambda}f(x)u+{\mid}u{\mid}^4u,&&\text{in }{\mathbb{R}}^3,\\-{\Delta}{\phi}=k(x)u^2,&&\text{in }{\mathbb{R}}^3,\end{array}$$ where b > 0, V (x), k(x) and f(x) are positive continuous smooth functions; 0 < λ < λ₁ and λ₁ is the first eigenvalue of the problem -∆u + V(x)u = λf(x)u in H. With the help of the constraint variational method, we obtain that the Schrödinger-Poisson-Kirchhoff type system possesses at least one ground state sign-changing solution for all b > 0 and 0 < λ < λ₁. Moreover, we prove that its energy is strictly larger than twice that of the ground state solutions of Nehari type.