Existence of positive solutions for a class of Kirchhoff type equations in $\mathbb{R}^3$

Ling  Ding; Shu-Ming Sun

doi:10.3934/dcdss.2016069

Article Contents

2016, Volume 9, Issue 6: 1663-1685. Doi: 10.3934/dcdss.2016069

This issue Previous Article Periodic solutions and homoclinic solutions for a Swift-Hohenberg equation with dispersion Next Article Global existence and uniqueness of the solution for the fractional Schrödinger-KdV-Burgers system

Existence of positive solutions for a class of Kirchhoff type equations in $\mathbb{R}^3$

Ling Ding^1, and
Shu-Ming Sun^2,

1.
School of Mathematics and Computer Science, Hubei University of Arts and Science, Xiangyang, Hubei 441053
2.
Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061

Received: June 30, 2015

Revised: August 31, 2016

Published: November 2016

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Abstract

The paper deals with the following equation of Kirchhoff type, \begin{align*} & -\left ( 1+b\left(\int_{\mathbb{R}^3}|\nabla u|^2dx\right)^r\, \right ) \Delta u+u=k(x)\left (|u|^{q-2}u+\theta g(u)\right )+\lambda h(x)u \end{align*} with $x\in \mathbb{R}^3$, where $u\in H^{1}(\mathbb{R}^3)$, $ b > 0, $ $0 < r < 2, q \in [2(r+1), 6) $, $\theta $ is a small constant, $\lambda$ is a parameter, and a weight function $h (x) \geq 0$. It is known that the linear operator $-\Delta u+u-\lambda h(x)u$ is coercive if $0<\lambda<\lambda_1(h)$ and is non-coercive if $\lambda>\lambda_1(h)$, where $\lambda_1(h)$ is the first eigenvalue of the operator $-\Delta u +u $ with the weight $h(x)$. Under suitable conditions on the functions $k(x)$ and $g(s)$, it is shown that the equation has a positive solution for any $\lambda\in(0,\lambda_1(h))$ and two positive solutions for $\lambda\in(\lambda_1(h), \lambda_1(h) + \tilde \delta )$ with $\tilde \delta > 0$ small. The conditions imposed on $k(x) $ and $g(s)$ are much weaker than those used before, thereby generalizing several existing results on the existence of positive solutions for this type of Kirchhoff equations.

Keywords:

Kirchhoff equations,
concentration compactness principle.

Mathematics Subject Classification: Primary: 35J20, 35B38; Secondary: 35B09.

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