American Institute of Mathematical Sciences

In this article, a sample of Norman Dancer's published works are briefly described, to give the reader a taste of his deep and important research on nonlinear functional analysis, nonlinear ODE and PDE problems, and dynamical systems. The sample covers a variety of topics where Norman Dancer has made remarkable contributions.

The author takes this opportunity to express his deep admiration of the work of Professor Norman Dancer, and to thank him for the kind help to the development of the author's career, which has been greatly influenced by him and his work. 

In a number of cases we calculate the sum of the degrees of the small positive solutions of the Gross-Pitaevskii system when the interaction is strong.

We prove that second order Hamiltonian systems \begin{document}$ -\ddot{u} = V_{u}(t,u) $\end{document} with a potential \begin{document}$ V\colon \mathbb{R} \times \mathbb{R} ^N\to \mathbb{R} $\end{document} of class \begin{document}$ C^1 $\end{document}, periodic in time and superquadratic at infinity with respect to the space variable have subharmonic solutions. Our intention is to generalise a result on subharmonics for Hamiltonian systems with a potential satisfying the global Ambrosetti-Rabinowitz condition from [14]. Indeed, we weaken the latter condition in a neighbourhood of \begin{document}$ 0\in \mathbb{R} ^N $\end{document}. We will also discuss when subharmonics pass to a nontrivial homoclinic orbit.

The standard way to study a compound singularity is to decompose it into the simpler ones using either blow up techniques or appropriate deformations. Among deformations, one distinguishes between miniversal deformations (related to deformations of a basis of the local algebra of singularity) and good deformations (one-parameter deformations with simple singularities coalescing into a multiple one). In concrete settings, explicit construction of a good deformation is an art rather than a science. In this paper, we discuss some cases important from the application viewpoint when explicit good deformations can be constructed and effectively used. Our applications include: (a) an \begin{document}$ n $\end{document}-dimensional Euler-Jacobi formula with simple and double roots, and (b) a simple approach to the known classification of phase portraits of planar differential systems around linearly non-zero equilibrium.

For a bounded domain \begin{document}$ \Omega\subset\mathbb{R}^n $\end{document} let \begin{document}$ H_\Omega:\Omega\times\Omega\to\mathbb{R} $\end{document} be the regular part of the Dirichlet Green function for the Laplace operator. Given a fixed arbitrary \begin{document}$ {\mathcal C}^2 $\end{document} function \begin{document}$ f:{\mathcal D}\to\mathbb{R} $\end{document}, defined on an open subset \begin{document}$ {\mathcal D}\subset\mathbb{R}^{nN} $\end{document}, and fixed coefficients \begin{document}$ \lambda_1,\dots,\lambda_N\in\mathbb{R}\setminus\{0\} $\end{document} we consider the function \begin{document}$ f_\Omega:{\mathcal D}\cap\Omega^N\to\mathbb{R} $\end{document} defined as

\begin{document}$ f_\Omega(x_1,\dots,x_N) = f(x_1,\dots,x_N) - \sum\limits_{j,k = 1}^N \lambda_j\lambda_k H_\Omega(x_j,x_k). $\end{document}

We prove that \begin{document}$ f_\Omega $\end{document} is a Morse function for most domains \begin{document}$ \Omega $\end{document} of class \begin{document}$ {\mathcal C}^{m+2,\alpha} $\end{document}, any \begin{document}$ m\ge0 $\end{document}, \begin{document}$ 0<\alpha<1 $\end{document}. This applies in particular to the Robin function \begin{document}$ h:\Omega\to\mathbb{R} $\end{document}, \begin{document}$ h(x) = H_\Omega(x,x) $\end{document}, and to the Kirchhoff-Routh path function where \begin{document}$ \Omega\subset\mathbb{R}^2 $\end{document}, \begin{document}$ {\mathcal D} = \{x\in\mathbb{R}^{2N}: x_j\ne x_k \; \text{for }\; j\ne k \} $\end{document}, and

\begin{document}$ f(x_1,\dots,x_N) = - \frac{1}{2\pi}\sum\limits_{{j,k = 1}\atop{j\ne k}}^N\lambda_j\lambda_k\log|x_j-x_k|. $\end{document}

In this paper, we investigate nonlinear periodic vibrations of a group of particles with a planar dihedral configuration governed by the Lennard-Jones and Coulomb forces. Using the gradient equivariant degree, we provide a full topological classification of the periodic solutions with both temporal and spatial symmetries. In the process, we provide general formulae for the spectrum of the linearized system of equations describing the above configuration, which allows us to obtain the critical frequencies of the particles' motions. The obtained frequencies represent the set of all critical periods for small amplitude periodic solutions emerging from a given stationary symmetric orbit of solutions.

In this paper, we consider the following coupled elliptic system

\begin{document}$ \begin{equation} \left\{ \begin{array}{ll} -\Delta u+\lambda_1 u = \mu_1 u^3+\beta uv^2-\gamma v &\text{in } \mathbb{R}^N, \\ -\Delta v+\lambda_2 v = \mu_2 v^3+\beta vu^2-\gamma u &\text{in } \mathbb{R}^N, \\ u(x), v(x)\rightarrow 0 \text{ as } \vert x\vert\rightarrow+\infty. \end{array} \right.\nonumber \end{equation} $\end{document}

Under symmetric assumptions \begin{document}$ \lambda_1 = \lambda_2, \mu_1 = \mu_2 $\end{document}, we determine the number of \begin{document}$ \gamma $\end{document}-bifurcations for each \begin{document}$ \beta\in(-1, +\infty) $\end{document}, and study the behavior of global \begin{document}$ \gamma $\end{document}-bifurcation branches in \begin{document}$ [-1, 0]\times H_r^1\left( \mathbb{R} ^N\right)\times H_r^1\left( \mathbb{R} ^N\right) $\end{document}. Moreover, several results for \begin{document}$ \gamma = 0 $\end{document}, such as priori bounds, are of independent interests, which are improvements of corresponding theorems in [6] and [35].

We study the following fractional Kirchhoff type equation:

\begin{document}$ \begin{equation*} \begin{array}{ll} \left \{ \begin{array}{ll} \Big(a+b\int_{ \mathbb{R} ^3}|(-\Delta)^\frac{s}{2}u|^2dx\Big)(-\Delta )^s u+V(x)u = f(u)+|u|^{2^*_s-2}u, \ x\in \mathbb{R} ^3, \\ u\in H^s( \mathbb{R} ^3), \end{array} \right . \end{array} \end{equation*} $\end{document}

where \begin{document}$ a, \ b>0 $\end{document} are constants, \begin{document}$ 2^*_s = \frac{6}{3-2s} $\end{document} with \begin{document}$ s\in(0, 1) $\end{document} is the critical Sobolev exponent in \begin{document}$ \mathbb{R} ^3 $\end{document}, \begin{document}$ V $\end{document} is a potential function on \begin{document}$ \mathbb{R} ^3 $\end{document}. Under some more general assumptions on \begin{document}$ f $\end{document} and \begin{document}$ V $\end{document}, we prove that the given problem admits a least energy solution by using a constrained minimization on Nehari-Pohozaev manifold and monotone method.

By relating the set of branch points \begin{document}$ \mathcal{B} (f) $\end{document} of a Fredholm mapping \begin{document}$ f $\end{document} to linearized bifurcation, we show, among other things, that under mild local assumptions at a single point, the set \begin{document}$ \mathcal B(f) $\end{document} is sufficiently large to separate the domain of the mapping. In the variational case, we will also provide estimates from below for the number of connected components of the complement of \begin{document}$ \mathcal B(f). $\end{document}

In this paper, we investigate the following quasilinear equation involving a Hardy potential:

\begin{document}$\begin{array}{l}\left\{ {\begin{array}{*{20}{c}}{ - \sum\limits_{i,j = 1}^N {{D_j}} ({a_{ij}}(u){D_i}u) + \frac{1}{2}\sum\limits_{i,j = 1}^N {{{a'}_{ij}}} (u){D_i}u{D_j}u - \frac{\mu }{{|x{|^2}}}u = au + |u{|^{{2^ * } - 2}}u}&{{\rm{in}}\;\Omega ,}\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{u = 0}&{{\rm{on}}\;\partial \Omega ,}\end{array}} \right.\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left( {\rm{P}} \right)\end{array}$\end{document}

where \begin{document}$ 2^ * = \frac{2N}{N-2} $\end{document} is the Sobolev critical exponent for the embedding of \begin{document}$ H_0^1(\Omega) $\end{document} into \begin{document}$ L^p(\Omega) $\end{document}, \begin{document}$ a>0 $\end{document} is a constant and \begin{document}$ \Omega\subset \mathbb{R}^N $\end{document} is an open bounded domain which contains the origin. We will prove that under some suitable assumptions on \begin{document}$ a_{ij} $\end{document}, when \begin{document}$ N\geq 7 $\end{document} and \begin{document}$ \mu\in[0,\mu^*) $\end{document} for some constant \begin{document}$ \mu^* $\end{document}, problem (P) admits an unbounded sequence of solutions. To achieve this goal, we perform the subcritical approximation and the regularization perturbation.

In this article we present a method of study of a global symmetry-breaking bifurcation of critical orbits of invariant functionals. As a topological tool we use the degree for equivariant gradient maps. We underline that many known results on bifurcations of non-radial solutions of elliptic PDE's from the families of radial ones are consequences of our theory.

We consider a Newton system which has a branch (surface) of neutrally stable periodic orbits. We discuss sufficient conditions which allow arbitrarily small delayed Pyragas control to make one selected cycle asymptotically stable. In the case of small amplitude periodic solutions we give conditions in terms of the asymptotic expansion of the right hand side, while in the case of larger cycles we frame the conditions in terms of the Floquet modes of the target orbit as a solution of the uncontrolled system.

In this paper, we investigate the following Choquard equation

\begin{document}$ \begin{equation*} -\Delta u = (I_\alpha*|u|^{f(x)})|u|^{f(x)-2}u \ \ \ \ {\rm in} \ \mathbb{R}^N, \end{equation*} $\end{document}

where \begin{document}$ N\geq 3 $\end{document}, \begin{document}$ \alpha\in (0,N) $\end{document} and \begin{document}$ I_\alpha $\end{document} is the Riesz potential. If

\begin{document}$ \begin{equation*} f(x) = \begin{cases} p, &x\in\Omega,\\ ( N+\alpha)/(N-2), &x\in\mathbb{R}^N \backslash\Omega, \end{cases} \end{equation*} $\end{document}

where \begin{document}$ 1< p <\frac{N+\alpha}{N-2} $\end{document} and \begin{document}$ \Omega\subset\mathbb{R}^N $\end{document} is a bounded set with nonempty, we obtain the existence of positive ground state solutions by using the Nehari manifold.

This paper is concerned with the nonlinear Dirac equation \begin{document}$ -i\sum_{k = 1}^{3}\alpha_{k}\partial_{k}u + [V(x)+a]\beta u + \omega u = f(x, u) $\end{document} in \begin{document}$ \mathbb{R}^3 $\end{document}, where \begin{document}$ V(x) $\end{document} and \begin{document}$ f(x, u) $\end{document} are periodic in \begin{document}$ x $\end{document}, \begin{document}$ f(x, u) $\end{document} is asymptotically linear and superlinear as \begin{document}$ |u|\rightarrow \infty $\end{document}. Under weaker assumptions on \begin{document}$ f $\end{document}, we obtain the existence of one nontrivial solution for the above equation.

In this paper, we consider a Leslie-Gower predator-prey model in one-dimensional environment. We study the asymptotic behavior of two species evolving in a domain with a free boundary. Sufficient conditions for spreading success and spreading failure are obtained. We also derive sharp criteria for spreading and vanishing of the two species. Finally, when spreading is successful, we show that the spreading speed is between the minimal speed of traveling wavefront solutions for the predator-prey model on the whole real line (without a free boundary) and an elliptic problem that follows from the original model.

We consider a 2\begin{document}$n$\end{document}th-order nonlinear difference equation containing both many advances and retardations with \begin{document}$\phi_c$\end{document}-Laplacian. Using the critical point theory, we obtain some new and concrete criteria for the existence and multiplicity of periodic and subharmonic solutions in the more general case of the nonlinearity.

The paper is devoted to traveling waves in FPU type particle chains assuming that each particle interacts with several neighbors on both sides. Making use of variational techniques, we prove that under natural assumptions there exist monotone traveling waves with periodic velocity profile (periodic waves) as well as waves with localized velocity profile (solitary waves). In fact, we obtain periodic waves by means of a suitable version of the Mountain Pass Theorem. Then we get solitary waves in the long wave length limit.

Inspired by Schaftingen [15], we develop a symmetric variational principle for the field equation involving a fractional Laplacians

\begin{document}$ \begin{equation*} \left\{ \begin{aligned} (-\Delta)^\alpha u+u& = f(u), x\in\mathbb{R}^N,\\ u(x)&\geq 0. \end{aligned} \right. \end{equation*} $\end{document}

As an application, we prove the existence of symmetric ground states in the fractional Sobolev space \begin{document}$ H^\alpha (\mathbb{R}^N) $\end{document}. These results improve some known ones in the literature. An example is also given to illustrate our results.

In the book "What is Mathematics?" Richard Courant and Herbert Robbins presented a solution of a Whitney's problem of an inverted pendulum on a railway carriage moving on a straight line. Since the appearance of the book in 1941 the solution was contested by several distinguished mathematicians. The first formal proof based on the idea of Courant and Robbins was published by Ivan Polekhin in 2014. Polekhin also proved a theorem on the existence of a periodic solution of the problem provided the movement of the carriage on the line is periodic. In the present paper we slightly improve the Polekhin's theorem by lowering the regularity class of the motion and we prove a theorem on the existence of a periodic solution if the carriage moves periodically on the plane.

In this paper, we study the following doubly coupled multicomponent system

\begin{document}$ \begin{equation*} \left\{\begin{array}{ll} -\Delta u_j + \lambda_ju_j+ \sum_{k\neq j}\gamma_{jk}u_k = \mu_ju_j^3+ u_j\sum_{k\neq j}\beta_{jk}u_k^2,\\ u_j(x)\geq0\ \ \hbox{and}\ \ u_j\in H_0^1(\Omega), \end{array} \right. \end{equation*} $\end{document}

where \begin{document}$ \Omega\subset \mathbb{R} ^N $\end{document} and \begin{document}$ N = 2,3 $\end{document}; \begin{document}$ \lambda_j, \gamma_{jk} = \gamma_{kj}, \mu_j, \beta_{jk} = \beta_{kj} $\end{document} are constants, \begin{document}$ j, k = 1, 2, ..., n $\end{document}, \begin{document}$ n\geq 2 $\end{document}. We prove some existence and nonexistence results for positive solutions of this system. If the system is fully symmetric, i.e. \begin{document}$ \lambda_j\equiv\lambda, \gamma_{jk}\equiv\gamma, \mu_j\equiv\mu, \beta_{jk}\equiv\beta $\end{document}, we study the multiplicity and bifurcation phenomena of positive solution.

Consider the second-order Hamiltonian system

\begin{document}$ \ddot{u}-L(t)u+\nabla W(t, u) = 0, $\end{document}

where \begin{document}$ t\in {\mathbb{R}}, u\in {\mathbb{R}}^{N} $\end{document}, \begin{document}$ L: \mathbb{R}\rightarrow {\mathbb{R}}^{N\times N} $\end{document} and \begin{document}$ W: {\mathbb{R}}\times {\mathbb{R}}^{N}\rightarrow {\mathbb{R}} $\end{document}. We mainly study the case when both \begin{document}$ L $\end{document} and \begin{document}$ W $\end{document} are periodic in \begin{document}$ t $\end{document} and \begin{document}$ 0 $\end{document} belongs to a spectral gap of \begin{document}$ \sigma\left(-\frac{d^2}{dt^2} +L\right) $\end{document}. We prove that the above system possesses a ground state homoclinic solution under assumptions which are weaker than the ones known in the literature.

Increasing experimental evidences suggest that cell phenotypic variation often depends on the accumulation of some special proteins. Recently, a lot of studies have shown that the complexity of promoter architecture plays a major role in regulating transcription and controlling expression dynamics and further phenotype. One unanswered question is why the organism chooses such a complex promoter architecture and how the promoter architecture affects the timing of proteins amount up to a given threshold. To address this issue, we study the effect of promoter architecture on the first-passage time (FPT) by formulating a multi-state gene model, that may reflect the complexity of promoter architecture. We derive analytical formulae for FPT moments in each case of irreversible promoter and reversible promoter regulation, which is the first time to give these analytical results in the existing literature. We show that the mean and noise of FPT increase with the state number of promoter architecture if the mean residence time at \begin{document}$ off$\end{document} states is not fixed. Inversely, if the mean residence time at \begin{document}$ off$\end{document} states is fixed, then complex promoter architecture will not vary the mean of FPT but will tend to decrease the noise of FPT. Our results show that, in the same inactive promoter states, the noise of FPT with promoters in irreversible case is always less than that in reversible case. In conclusion, our results reveal the effect of the promoter architecture on FPT and enhance understanding of the regulation mechanism of gene expression.

A stage-structured predator-prey model with prey refuge is considered. Using the geometric stability switch criteria, we establish stability of the positive equilibrium. Stability and direction of periodic solutions arising from Hopf bifurcations are obtained by using the normal form theory and center manifold argument. Numerical simulations confirm the above theoretical results.