Electron. J. Diff. Equ., Vol. 2016 (2016), No. 35, pp. 1-22.

Existence of two positive solutions for indefinite Kirchhoff equations in R^3

Ling Ding, Yi-Jie Meng, Shi-Wu Xiao, Jin-Ling Zhang

Abstract:
In this article we study the Kirchhoff type equation
$$\displaylines{
 -\Big(1+b\int_{\mathbb{R}^3}|\nabla u|^2dx\Big)\Delta u+u
 =k(x)f(u)+\lambda h(x)u,\quad x\in \mathbb{R}^3, \cr
 u\in H^{1}(\mathbb{R}^3),
 }$$
involving a linear part $-\Delta u+u-\lambda h(x)u$ which is coercive if $0<\lambda<\lambda_1(h)$ and is noncoercive if $\lambda>\lambda_1(h)$, a nonlocal nonlinear term $-b\int_{\mathbb{R}^3}|\nabla u|^2dx\Delta u$ and a sign-changing nonlinearity of the form $k(x)f(s)$, where $ b>0$, $\lambda>0$ is a real parameter and $\lambda_1(h)$ is the first eigenvalue of $-\Delta u+u=\lambda h(x)u$. Under suitable assumptions on f and h, we obtain positives solution for $\lambda\in(0,\lambda_1(h))$ and two positive solutions with a condition on k.

Submitted August 20, 2015. Published January 25, 2016.
Math Subject Classifications: 35J60, 35B38, 35A15.
Key Words: Indefinite Kirchhoff equation; concentration compactness lemma; (PS) condition; Ekeland's variational principle.

Show me the PDF file (324 KB), TEX file for this article.

Ling Ding
School of Mathematics and Computer Science
Hubei University of Arts and Science
Hubei 441053, China
email: dingling1975@qq.com, dingling19750118@163.com
Yi-Jie Meng
School of Mathematics and Computer Science
Hubei University of Arts and Science
Hubei 441053, China
email: 245330581@qq.com
Shi-Wu Xiao
School of Mathematics and Computer Science
Hubei University of Arts and Science
Hubei 441053, China
email: xshiwu@sina.com
Jin-Ling Zhang
School of Mathematics and Computer Science
Hubei University of Arts and Science
Hubei 441053, China
email: jinling48@163.com

Return to the EJDE web page