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This paper is a sequel to Byeon et al. (J Math Pures Appl 106(9):477–511, 2016) concerning the asymptotic behavior of positive least energy vector solutions to nonlinear Schrodinger systems with mixed couplings that arise from models in Bose–Einstein condensates and nonlinear optics. In [8] we treated homogeneous Dirichlet boundary condition. In the current paper we investigate the case of homogeneous Neumann boundary condition. We show that due to mixed attractive and repulsive interactions the least energy solutions exhibit component-wise pattern formations, in particular, co-existence of partial synchronization and segregation. We employ multiple scalings to carry out a refined asymptotic analysis of convergence to a multiple scaled limiting system.

In this paper, we are interested in standing waves with a vortex for the nonlinear Chern–Simons–Schrodinger 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.

For $k =1,\cdots,K,$ let $M_k$ be a $q_k$-dimensional smooth compact framed manifold in $R^N$ with $q_k \in \{1,\cdots,N-1\} $. We consider the equation $-\varepsilon^2\Delta u + V(x)u - u^p = 0$ in $R^N$ where for each $k \in \{1,\cdots,K\}$ and some $m_k > 0,$ $V(x)=|\textrm{dist}(x,M_k)|^{m_k}+O(|\textrm{dist}(x,M_k)|^{m_k+1})$ as $\textrm{dist}(x,M_k) \to 0 $. For a sequence of $\varepsilon$ converging to zero, we will find a positive solution $u_{\varepsilon}$ of the equation which concentrates on $M_1\cup \dots \cup M_K$.

We consider the singularly perturbed nonlinear elliptic problem \begin{eqnarray*} \varepsilon^2 \Delta v - V(x)v + f(v) =0, v > 0, \lim_{|x|\to \infty} v(x) = 0. \end{eqnarray*} Under almost optimal conditions for the potential $V$ and the nonlinearity $f$, we establish the existence of single-peak solutions whose peak points converge to local minimum points of $V$ as $\varepsilon \to 0$. Moreover, we exhibit a threshold on the condition of $V$ at infinity between existence and nonexistence of solutions.

We study standing waves for nonlinear Schrodinger 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 A μ 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.

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.

We study semiclassical states of nonlinear Schrodinger 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.

For m > 0 and p ∈ (1, (N + 2)/(N − 2)), we show the uniqueness and a linearized non-degeneracy of solutions for the following problem

For N = 1,2, we consider singularly perturbed elliptic equations ϵ2Δ u − V(x) u + f(u)= 0, u(x)> 0 on R N , lim|x|→∞ u(x)= 0. For small ϵ > 0, we show the existence of a localized bound state solut...

For singularly perturbed Schrodinger equations with decaying potentials at infinity we construct semiclassical states of a critical frequency concentrating on spheres near zeroes of the potentials. The results generalize some recent work of Ambrosetti-Malchiodi-Ni[3] which gives solutions concentrating on spheres where the potential is positive. The solutions we obtain exhibit different behaviors from the ones given in [3].

For elliptic equations of the form $\Delta u -V(\varepsilon x) u + f(u)=0, x\in {\bf R}^N$ , where the potential V satisfies $\liminf_{\vert x\vert\to \infty} V(x) > \inf_{{\bf R}^N} V(x) =0$ , we develop a new variational approach to construct localized bound state solutions concentrating at an isolated component of the local minimum of V where the minimum value of V can be positive or zero. These solutions give rise to standing wave solutions having a critical frequency for the corresponding nonlinear Schrodinger equations. Our method allows a fairly general class of nonlinearity f(u) including ones without any growth restrictions at large.

This paper is concerned with the existence and qualitative property of standing wave solutions \(\) for the nonlinear Schrodinger equation \(\) with E being a critical frequency in the sense that \(\). We show that there exists a standing wave which is trapped in a neighbourhood of isolated minimum points of V and whose amplitude goes to 0 as \(\). Moreover, depending upon the local behaviour of the potential function V(x) near the minimum points, the limiting profile of the standing-wave solutions will be shown to exhibit quite different characteristic features. This is in striking contrast with the non-critical frequency case \(\) which has been extensively studied in recent years.

Erratum: Arch. Rational Mech. Anal. DOI 10.1007/s00205-006-0019-3Through answering to some questions from Joao Marcos Bezzerra do Oon the paper above, we found some mistakes in the paper. We thank him forhis interest and careful reading. Here we correct these mistakes.First at the beginning of the proof of Proposition 4, 14-17 line on page193, the correct version is:By compactness of S

For elliptic equations e2Δu − V(x) u + f(u) = 0, x ∈ RN, N ≧ 3, we develop a new variational approach to construct localized positive solutions which concentrate at an isolated component of positive local minimum points of V, as e → 0, under conditions on f which we believe to be almost optimal.

Let Ω be a bounded domain in R N with the boundary ∂ Ω ∈ C 3 . We consider the following singularly perturbed nonlinear elliptic problem on Ω, e 2 Δ v − v + f ( v ) = 0 , v > 0 on Ω , ∂ v ∂ ν = 0 on ∂ Ω , where ν is the exterior normal to ∂Ω and the nonlinearity f is of subcritical growth. It has been known that under Berestycki and Lions conditions for f ∈ C 1 ( R ) and N ⩾ 3 , there exists a solution v e of the problem which develops a spike layer near a local maximum point of the mean curvature H on ∂Ω for small e > 0 . In this paper, we extend the previous result for f ∈ C 0 ( R ) and N ⩾ 2 .

We consider singularly perturbed elliptic equations $\varepsilon^2\Delta u - V(x) u + f(u)=0, x\in R^N, N \ge 3.$ For small $\varepsilon > 0,$ we glue together localized bound state solutions concentrating at isolated components of positive local minimum of $V$ under conditions on $f$ we believe to be almost optimal.