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In this paper, we are interested in standing waves with a vortex for the nonlinear Chern–Simons–Schrödinger equations (CSS for short). We study the existence and the nonexistence of standing waves when a constant λ > 0 , representing the strength of the interaction potential, varies. We prove every standing wave is trivial if λ ∈ ( 0 , 1 ) , every standing wave is gauge equivalent to a solution of the first order self-dual system of CSS if λ = 1 and for every positive integer N , there is a nontrivial standing wave with a vortex point of order N if λ > 1 . We also provide some classes of interaction potentials under which the nonexistence of standing waves and the existence of a standing wave with a vortex point of order N are proved. [ABSTRACT FROM AUTHOR]

For $$N \le 3$$ and $$\beta > 0,$$ we consider the following singularly perturbed elliptic system There are an enormous number of results for localized solutions of singularly perturbed scalar problems using variational methods or finite dimensional reduction methods. However, there exist no general existence results of localized solutions for elliptic systems. We present some such results here. In the first, by a mini-max characterization for a limiting problem, for small $$\varepsilon > 0,$$ we show the existence of one bump solutions with a common concentration point of $$u_1,u_2$$ in a domain O when certain conditions for the limiting problem are satisfied. Typical examples of potentials $$W_1,W_2$$ satisfying the condition are the following: (1) $$W_1,W_2$$ have a common non-degenerate critical point in O which share the same stable, unstable directions; (2) for the outnormal n on $$\partial O$$ , $$\frac{\partial W_1}{\partial n} > 0, \frac{\partial W_2}{\partial n} > 0$$ or $$\frac{\partial W_1}{\partial n} < 0, \frac{\partial W_2}{\partial n} < 0$$ on $$\partial O;$$ (3) $$\max _{x \in O}W_i(x) >> \max _{x \in \partial O}W_i(x)$$ for $$i =1,2.$$ We also give some nonexistence results for some potentials $$W_1,W_2$$ , not satisfying these conditions, but each $$W_1,W_2$$ having a structurally stable critical point in O. [ABSTRACT FROM AUTHOR]

Abstract: We study standing waves for nonlinear Schrödinger equations with the gauge field. Some existence results of standing waves are established by applying variational methods to the functional which is obtained by representing the gauge field in terms of complex scalar field ϕ. We also show that there exists no standing wave for certain range of parameters by establishing a new inequality of Sobolev type. [Copyright &y& Elsevier]

Abstract: We glue together standing wave solutions concentrating around critical points of the potential V with different energy scales. We devise a hybrid method using simultaneously a Lyapunov–Schmidt reduction method and a variational method to glue together standing waves concentrating on local minimum points which possibly have no corresponding limiting equations and those concentrating on general critical points which converge to solutions of corresponding limiting problems satisfying a non-degeneracy condition. [Copyright &y& Elsevier]

Abstract: We study semiclassical states of nonlinear Schrödinger equations with anisotropic type potentials which may exhibit a combination of vanishing and singularity while allowing decays and unboundedness at infinity. We give existence of spike type standing waves concentrating at the singularities of the potentials. [Copyright &y& Elsevier]

For elliptic equations of the form ε² Δu - V(x)u + up = 0, x ∈ RN, where the potential V satisfies lim inf¦x¦→∞ V(x) > infRN V(x) = 0, we prove the existence of new kinds of solutions, corresponding to semi-classical standing waves for nonlinear Schrödinger equations, with several local maximum points whose local maximum values are of different scales with respect to ε → 0. [ABSTRACT FROM AUTHOR]

A correction to the article "Standing Waves for Nonlinear Schrödinger Equations With a General Nonlinearity," by Jaeyoung Byeon and Louis Jeanjean is presented.