Multiple solutions for Schrodinger-Poisson systems with sign-changing potential and critical nonlinearity - Liuyang Shao; Haibo Chen (Electronic Journal of Differential Equations, 2016)
In this article, we study the Schr\"odinger-Poisson system $$\displaylines{ -Δ u+V(x)u+k(x)φ(x)u=h_1(x)|u|^{4}u+μ h_{2}(x)u+h_3(x), \quad\text{in } \mathbb{R}^3, \cr -Δ φ(x)=k(x)u^2 , \quad\text{in } \mathbb{R}^3, }$$ where $h_1(x),h_{2}(x),h_3(x), V(x)$ are allowed to be sign-changing and $μ>0$ is a parameter. Under some appropriate assumptions on V(x), we obtain the existence of two different solutions for the above system via variational methods.