Taiwanese Journal of Mathematics

Existence of Nonnegative Solutions for Fourth Order Elliptic Equations of Kirchhoff Type with General Subcritical Growth

Jianping Huang and Qi Zhang

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Abstract

This paper is dedicated to investigating the following fourth-order elliptic equation with Kirchhoff-type \[ \begin{cases} \displaystyle \Delta^{2} u - \left( a + b \int_{\mathbb{R}^{N}} |\nabla u|^{2} \, dx \right) \Delta u + cu = f(u) &\textrm{in $\mathbb{R}^{N}$}, \\ u \in H^{2}(\mathbb{R}^{N}), \end{cases} \] where $a \gt 0$, $b \geq 0$ and $c \gt 0$ are constants. By using cut-off functional and monotonicity tricks, we prove that the above problem has a positive solution. Our result cover the case where the nonlinearity satisfies asymptotically linear and superlinear condition at the infinity, which extend the results of related literatures.

Article information

Source
Taiwanese J. Math., Advance publication (2019), 16 pages.

Dates
First available in Project Euclid: 26 February 2019

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1551150032

Digital Object Identifier
doi:10.11650/tjm/190202

Subjects
Primary: 35J35: Variational methods for higher-order elliptic equations 35J50: Variational methods for elliptic systems 35J60: Nonlinear elliptic equations

Keywords
fourth order elliptic equations cut-off functional Pohožaev equality

Citation

Huang, Jianping; Zhang, Qi. Existence of Nonnegative Solutions for Fourth Order Elliptic Equations of Kirchhoff Type with General Subcritical Growth. Taiwanese J. Math., advance publication, 26 February 2019. doi:10.11650/tjm/190202. https://projecteuclid.org/euclid.twjm/1551150032


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