Abstract
Abstract In this paper we consider the nonlinear Chern-Simons-Schrödinger equations with general nonlinearity $$\begin{array}{} \displaystyle -{\it\Delta} u+\lambda V(|x|)u+\left(\frac{h^2(|x|)}{|x|^2}+\int\limits^{\infty}_{|x|}\frac{h(s)}{s}u^2(s)ds\right)u=f(u),\,\, x\in\mathbb R^2, \end{array}$$ where λ > 0, V is an external potential and $$\begin{array}{} \displaystyle h(s)=\frac{1}{2}\int\limits^s_0ru^2(r)dr=\frac{1}{4\pi}\int\limits_{B_s}u^2(x)dx \end{array}$$ is the so-called Chern-Simons term. Assuming that the external potential V is nonnegative continuous function with a potential well Ω := int V–1(0) consisting of k + 1 disjoint components Ω0, Ω1, Ω2 ⋯, Ωk, and the nonlinearity f has a general subcritical growth condition, we are able to establish the existence of sign-changing multi-bump solutions by using variational methods. Moreover, the concentration behavior of solutions as λ → +∞ are also considered.
Highlights
Introduction and main resultsIn this paper we are interested in the following nonlinear Schrödinger system with the gauge eld iD φ + (D D+ D D )φ + g(φ) = ∂ A −∂ A = −Im(φ D φ),∂ A − ∂ A = Im(φ D φ), (1.1) ∂A =−
→ C is the complex scalar eld, Aκ : R + → R is the gauge eld and Dκ = ∂κ + iAκ is the covariant derivative for κ =
The two-dimensional Chern-Simons-Schrödinger equations is a nonrelativistic quantum model describing the dynamics of a large number of particles in the plane, which interact both
Summary
In this paper we are interested in the following nonlinear Schrödinger system with the gauge eld iD φ + → C is the complex scalar eld, Aκ : R + → R is the gauge eld and Dκ = ∂κ + iAκ is the covariant derivative for κ = , ,. This model (1.1) was rst proposed and studied in [22,23,24], and is sometimes called the Chern-. The two-dimensional Chern-Simons-Schrödinger equations is a nonrelativistic quantum model describing the dynamics of a large number of particles in the plane, which interact both
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