Sign-changing solutions to a gauged nonlinear Schrödinger equation

Abstract

We study the existence and asymptotic behavior of the least energy sign-changing solutions to a gauged nonlinear Schrödinger equation{−Δu+ωu+λ(h2(|x|)|x|2+∫|x|+∞h(s)su2(s)ds)u=|u|p−2u, x∈R2,u∈Hr1(R2), where ω,λ>0, p>6 andh(s)=12∫0sru2(r)dr. Combining constraint minimization method and quantitative deformation lemma, we prove that the problem possesses at least one least energy sign-changing solution uλ, which changes sign exactly once. Moreover, we show that the energy of uλ is strictly larger than two times of the ground state energy. Finally, the asymptotic behavior of uλ as λ↘0 is also analyzed.