Existence of least energy nodal solution for Kirchhoff–Schrödinger–Poisson system with potential vanishing

• Main Authors: Jin-Long Zhang, Da-Bin Wang
• Format: Article
• Language: English
• Published: SpringerOpen 2020-06-01
• Series: Boundary Value Problems
• Subjects: Potential vanishing; Nehari manifold; Least energy nodal solution
• Online Access: http://link.springer.com/article/10.1186/s13661-020-01408-2

Abstract This paper deals with the following Kirchhoff–Schrödinger–Poisson system: { − ( a + b ∫ R 3 | ∇ u | 2 d x ) Δ u + V ( x ) u + ϕ u = K ( x ) f ( u ) in  R 3 , − Δ ϕ = u 2 in  R 3 , $$ \textstyle\begin{cases} -(a+b\int _{\mathbb{R}^{3}} \vert \nabla u \vert ^{2}\,dx)\Delta u+V(x)u+\phi u=K(x)f(u)&\text{in } \mathbb{R}^{3}, \\ -\Delta \phi =u^{2}&\text{in } \mathbb{R}^{3}, \end{cases} $$ where a, b are positive constants, K ( x ) $K(x)$ , V ( x ) $V(x)$ are positive continuous functions vanishing at infinity, and f ( u ) $f(u)$ is a continuous function. Using the Nehari manifold and variational methods, we prove that this problem has a least energy nodal solution. Furthermore, if f is an odd function, then we obtain that the equation has infinitely many nontrivial solutions.