Volume 30, Issue 2
Existence Theorem for a Class of Nonlinear Fourth-order Schrödinger-Kirchhoff-Type Equations.

J. Part. Diff. Eq., 30 (2017), pp. 146-164.

Published online: 2017-05

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• Abstract

This paper is concerned with the existence of nontrivial solutions for the following fourth-order equations of Kirchhoff type

\begin{equation*}\begin{cases}\Delta^{2}u-\left(a+b\displaystyle\int_{{\mathbb{R}}^N }|\nabla{u}|^2{\rm d}x\right)\Delta{u}+\lambda V(x)u=f(x,u),\quad x\in\mathbb{R}^N ,\\u\in{H^2({\mathbb{R}}^N)},\end{cases}\end{equation*}

where $a,b$ are positive constants, $\lambda \geq 1$ is a parameter, and the nonlinearity $f$ is either superlinear or sublinear  at infinity in $u$. With the help of the variational methods, we obtain the existence and multiplicity results in the working spaces.

• Keywords

Fourth-order elliptic equations symmetric mountain pass theorem Morse theory

52B10 65D18 68U05 68U07

tangshiqiang@whu.edu.cn (Shiqiang Tang)

whuchenpeng@163.com (Peng Chen)

xcliu@whu.edu.cn (Xiaochun Liu)

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@Article{JPDE-30-146, author = {Tang , Shiqiang and Chen , Peng and Liu , Xiaochun}, title = {Existence Theorem for a Class of Nonlinear Fourth-order Schrödinger-Kirchhoff-Type Equations.}, journal = {Journal of Partial Differential Equations}, year = {2017}, volume = {30}, number = {2}, pages = {146--164}, abstract = {

This paper is concerned with the existence of nontrivial solutions for the following fourth-order equations of Kirchhoff type

\begin{equation*}\begin{cases}\Delta^{2}u-\left(a+b\displaystyle\int_{{\mathbb{R}}^N }|\nabla{u}|^2{\rm d}x\right)\Delta{u}+\lambda V(x)u=f(x,u),\quad x\in\mathbb{R}^N ,\\u\in{H^2({\mathbb{R}}^N)},\end{cases}\end{equation*}

where $a,b$ are positive constants, $\lambda \geq 1$ is a parameter, and the nonlinearity $f$ is either superlinear or sublinear  at infinity in $u$. With the help of the variational methods, we obtain the existence and multiplicity results in the working spaces.

}, issn = {2079-732X}, doi = {https://doi.org/10.4208/jpde.v30.n2.3}, url = {http://global-sci.org/intro/article_detail/jpde/10003.html} }
TY - JOUR T1 - Existence Theorem for a Class of Nonlinear Fourth-order Schrödinger-Kirchhoff-Type Equations. AU - Tang , Shiqiang AU - Chen , Peng AU - Liu , Xiaochun JO - Journal of Partial Differential Equations VL - 2 SP - 146 EP - 164 PY - 2017 DA - 2017/05 SN - 30 DO - http://doi.org/10.4208/jpde.v30.n2.3 UR - https://global-sci.org/intro/article_detail/jpde/10003.html KW - Fourth-order elliptic equations KW - symmetric mountain pass theorem KW - Morse theory AB -

This paper is concerned with the existence of nontrivial solutions for the following fourth-order equations of Kirchhoff type

\begin{equation*}\begin{cases}\Delta^{2}u-\left(a+b\displaystyle\int_{{\mathbb{R}}^N }|\nabla{u}|^2{\rm d}x\right)\Delta{u}+\lambda V(x)u=f(x,u),\quad x\in\mathbb{R}^N ,\\u\in{H^2({\mathbb{R}}^N)},\end{cases}\end{equation*}

where $a,b$ are positive constants, $\lambda \geq 1$ is a parameter, and the nonlinearity $f$ is either superlinear or sublinear  at infinity in $u$. With the help of the variational methods, we obtain the existence and multiplicity results in the working spaces.

Shiqiang Tang, Peng Chen & Xiaochun Liu. (2019). Existence Theorem for a Class of Nonlinear Fourth-order Schrödinger-Kirchhoff-Type Equations.. Journal of Partial Differential Equations. 30 (2). 146-164. doi:10.4208/jpde.v30.n2.3
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