## Abstract and Applied Analysis

### Positive Solutions for a Nonhomogeneous Kirchhoff Equation with the Asymptotical Nonlinearity in ${R}^{3}$

#### Abstract

We study the following nonhomogeneous Kirchhoff equation: $-(a+b{\int }_{{R}^{\mathrm{3}}}\mathrm{‍}|\nabla u{|}^{\mathrm{2}}dx)\mathrm{\Delta }u+u=k(x)f(u)+h(x), x\in {R}^{\mathrm{3}}, u\in {H}^{\mathrm{1}}({R}^{\mathrm{3}}), u>\mathrm{0}, x\in {R}^{\mathrm{3}}$, where $f$ is asymptotically linear with respect to $t$ at infinity. Under appropriate assumptions on $k,f$, and $h$, existence of two positive solutions is proved by using the Ekeland's variational principle and the Mountain Pass Theorem in critical point theory.

#### Article information

Source
Abstr. Appl. Anal., Volume 2014 (2014), Article ID 710949, 10 pages.

Dates
First available in Project Euclid: 2 October 2014

https://projecteuclid.org/euclid.aaa/1412273194

Digital Object Identifier
doi:10.1155/2014/710949

Mathematical Reviews number (MathSciNet)
MR3176766

Zentralblatt MATH identifier
07022928

#### Citation

Ding, Ling; Li, Lin; Zhang, Jin-Ling. Positive Solutions for a Nonhomogeneous Kirchhoff Equation with the Asymptotical Nonlinearity in ${R}^{3}$. Abstr. Appl. Anal. 2014 (2014), Article ID 710949, 10 pages. doi:10.1155/2014/710949. https://projecteuclid.org/euclid.aaa/1412273194

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