Nonlocal Kirchhoff superlinear equations with indefinite nonlinearity and lack of compactness

We study the following Kirchhoff equation $$- \left(1 + b \int_{\mathbb{R}^3} |\nabla u|^2 dx \right) \Delta u + V(x) u = f(x,u), \ x \in \mathbb{R}^3.$$ A special feature of this paper is that the nonlinearity $f$ and the potential $V$ are indefinite, hence sign-changing. Under some appropriate assumptions on $V$ and $f$, we prove the existence of two different solutions of the equation via the Ekeland variational principle and Mountain Pass Theorem

On a Kirchhoff type problems with potential well and indefinite potential

In this paper, we study the following Kirchhoff type problem:% \left\{\aligned&-\bigg(\alpha\int_{\bbr^3}|\nabla u|^2dx+1\bigg)\Delta u+(\lambda a(x)+a_0)u=|u|^{p-2}u&\text{ in }\bbr^3,\\% &u\in\h,\endaligned\right.\eqno{(\mathcal{P}_{\alpha,\lambda})}% where $4<p<6$, $\alpha$ and $\lambda$ are two positive parameters, a_0\in\bbr is a (possibly negative) constant and $a(x)\geq0$ is the potential well. By the variational method, we investigate the existence of nontrivial solutions to $(\mathcal{P}_{\alpha,\lambda})$. To our best knowledge, it is the first time that the nontrivial solution of the Kirchhoff type problem is found in the indefinite case. We also obtain the concentration behaviors of the solutions as $\lambda\to+\infty$.Comment: 1