2013 The Existence of Multiple Solutions for Nonhomogeneous Kirchhoff Type Equations in ${ℝ}^{3}$
Qi Zhang, Xiaoli Zhu
Abstr. Appl. Anal. 2013: 1-5 (2013). DOI: 10.1155/2013/806865

## 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 $|h{|}_{\mathrm{2}}<{m}_{p}$.

## Citation

Qi Zhang. Xiaoli Zhu. "The Existence of Multiple Solutions for Nonhomogeneous Kirchhoff Type Equations in ${ℝ}^{3}$." Abstr. Appl. Anal. 2013 1 - 5, 2013. https://doi.org/10.1155/2013/806865

## Information

Published: 2013
First available in Project Euclid: 27 February 2014

MathSciNet: MR3134160
Digital Object Identifier: 10.1155/2013/806865 