Ground state radial sign-changing solutions for a gauged nonlinear Schrödinger equation involving critical growth

Abstract

We investigate the following gauged nonlinear Schrödinger equation      −∆u + ωu + λ h 2 u (|x|) |x| 2 + +∞ |x| hu(s) s u 2 (s)ds u = f (u) in R 2 , u ∈ H 1 r (R 2), where ω, λ > 0 and hu(s) = 1 2 s 0 ru 2 (r)dr. When f has exponential critical growth, by using the constrained minimization method and Trudinger-Moser inequality, it is proved that the equation has a ground state radial sign-changing solution u λ which changes sign exactly once. Moreover, the asymptotic behavior of u λ as λ → 0 is analyzed.

Existence of nontrivial solutions to Chern-Simons-Schrödinger system with indefinite potential

“…The existence and nonexistence results on nontrivial radial solutions have been obtained when g(u) = λ|u| q−2 u, λ > 0 and q > 2. Later, based on the work of [2], system (2) have been studied by many researchers in radial functions space H 1 r (R 2 ), and they obtained lots of results, see [2,3,7,15,14,18,19,17,26,27,28,32] and references therein. After these works, mathematicians also began to consider system (2) in H 1 (R 2 ).…”