## Abstract and Applied Analysis

### The Existence of Multiple Solutions for Nonhomogeneous Kirchhoff Type Equations in ${ℝ}^{3}$

#### Abstract

We are concerned with the existence of multiple solutions to the nonhomogeneous Kirchhoff type equation $-\left(a+b{\int }_{{ℝ}^{\mathrm{3}}}\mathrm{‍}|\nabla u{|}^{\mathrm{2}}\right)\mathrm{\Delta }u+u=|u{|}^{p-\mathrm{1}}u+h\left(x\right)\mathrm{}\mathrm{}\text{\hspace\left\{0.17em\right\}\hspace\left\{0.17em\right\}in\hspace\left\{0.17em\right\}\hspace\left\{0.17em\right\}}\mathrm{}\mathrm{}\mathrm{}\mathrm{}{ℝ}^{\mathrm{3}},$ where $a, b$ are positive constants, $p\in \left(\mathrm{1,5}\right), \mathrm{0}⩽h\left(x\right)=h\left(|x|\right)\in {C}^{\mathrm{1}}\left({ℝ}^{\mathrm{3}}\right)\cap {L}^{\mathrm{2}}\left({ℝ}^{\mathrm{3}}\right)$, we can find a constant ${m}_{p}>\mathrm{0}$ such that for all $p\in \left(\mathrm{1,5}\right)$ the equation has at least two radial solutions provided $|h{|}_{\mathrm{2}}<{m}_{p}$.

#### Article information

Source
Abstr. Appl. Anal., Volume 2013 (2013), Article ID 806865, 5 pages.

Dates
First available in Project Euclid: 27 February 2014

https://projecteuclid.org/euclid.aaa/1393512172

Digital Object Identifier
doi:10.1155/2013/806865

Mathematical Reviews number (MathSciNet)
MR3134160

#### Citation

Zhang, Qi; Zhu, Xiaoli. The Existence of Multiple Solutions for Nonhomogeneous Kirchhoff Type Equations in ${ℝ}^{3}$. Abstr. Appl. Anal. 2013 (2013), Article ID 806865, 5 pages. doi:10.1155/2013/806865. https://projecteuclid.org/euclid.aaa/1393512172

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