Infinitely Many Solutions for Schrödinger–Poisson Systems and Schrödinger–Kirchhoff Equations

Abstract

By applying Clark’s theorem as altered by Liu and Wang and the truncation method, we obtain a sequence of solutions for a Schrödinger–Poisson system −Δu+V(x)u+ϕu=f(u)inR3,−Δϕ=u2inR3 with negative energy. A similar result is also obtained for the Schrödinger-Kirchhoff equation as follows:−1+∫RN∇u2Δu+V(x)u=f(u)u∈H1(RN).

Keywords

• Schrödinger equations
• Clark’s theorem
• truncation method
• Palais–Smale condition