| Summary: | 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.
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