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.

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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

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