Sign-changing solutions for the nonlinear Chern–Simons–Schrödinger equations

Abstract

In this paper, we study the existence and asymptotic behavior of least energy sign-changing solutions for the nonlinear Chern–Simons–Schrödinger equations −Δu+ωu+λh2(|x|)|x|2+∫|x|∞h(s)su2(s)dsu=f(u), x∈R2,u∈Hr1(R2), where ω, λ>0, f∈C(R,R) and h(s)=12∫0sru2(r)dr. Under suitable assumptions on f, we use some analytical skills and constraint minimization method to show that the above problem admits one least energy sign-changing solution uλ with precisely two nodal domains. Furthermore, we show that the energy of uλ is strictly larger than two times of the least energy, and present a convergence property of uλ as λ→0+. Finally, we also prove that the above results are valid for the Chern–Simons–Schrödinger equations with steep well potential.