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 Schrodinger equation { − Δ u + ω u + λ ( h 2 ( | x | ) | x | 2 + ∫ | x | + ∞ h ( s ) s u 2 ( s ) d s ) u = | u | p − 2 u , x ∈ R 2 , u ∈ H r 1 ( R 2 ) , where ω , λ > 0 , p > 6 and h ( s ) = 1 2 ∫ 0 s r u 2 ( r ) d r . Combining constraint minimization method and quantitative deformation lemma, we prove that the problem possesses at least one least energy sign-changing solution u λ , which changes sign exactly once. Moreover, we show that the energy of u λ is strictly larger than two times of the ground state energy. Finally, the asymptotic behavior of u λ as λ ↘ 0 is also analyzed.