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

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 $0\lambda_1(h)$, 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.

Keywords

• Indefinite Kirchhoff equation
• concentration compactness lemma
• (PS) condition
• Ekeland's variational principle