Related papers: Bifurcation in a multi-component system of nonline…
We develop numerical algorithms to approximate positive solutions of elliptic boundary value problems with superlinear subcritical nonlinearity on the boundary of the form $-\Delta u + u = 0$ in $\Omega$ with $\frac{\partial u}{\partial…
We investigate symmetry properties of positive solutions for fully nonlinear uniformly elliptic systems, such as $$ F_i \,(x,Du_i,D^2u_i) +f_i \,(x,u_1, \ldots , u_n,Du_i)=0, \;\; 1 \leq i \leq n, $$ in a bounded domain $\Omega$ in…
In this paper, we shall establish the unilateral global bifurcation result for a class of fourth-order eigenvalue problems with sign-changing weight. Under some natural hypotheses on perturbation function, we show that $(\mu_k^\nu,0)$ is a…
We introduce a system of two linearly coupled discrete nonlinear Schr\"{o}dinger equations (DNLSEs), with the coupling constant subject to a rapid temporal modulation. The model can be realized in bimodal Bose-Einstein condensates (BEC).…
In this series of papers, we investigate coupled systems arising in the study of two-component Bose--Einstein condensates, and we establish classification results for solutions of De Giorgi conjecture type. In the first paper of the series,…
We consider a nonlinear Dirichlet problem driven by a nonhomogeneous differential operator with a growth of order $(p-1)$ near $+\infty$ and with a reaction which has the competing effects of a parametric singular term and a…
We study the existence of fully nontrivial solutions to the system $$-\Delta u_i+ \lambda_iu_i = \sum\limits_{j=1}^\ell \beta_{ij}|u_j|^p|u_i|^{p-2}u_i\ \hbox{in}\ \Omega, \qquad i=1,\ldots,\ell,$$ in a bounded or unbounded domain $\Omega$…
We investigate the bifurcation structure of stationary states in a dipolar Bose-Einstein condensate located in an external PT-symmetric potential. The imaginary part of this external potential allows for the effective description of in- and…
In this paper we treat the behavior of Bose Einstein condensates in double square well potentials, both of equal and different depths. For even depth, symmetry preserving solutions to the relevant nonlinear Schr\"{o}dinger equation is…
We consider the bifurcation problem $u'' + \lambda u = N(u)$ with two point boundary conditions where $N(u)$ is a general nonlinear term which may also depend on the eigenvalue $\lambda$. We give a variational characterization of the…
We present new local and global dynamic bifurcation results for nonlinear evolution equations of the form $u_t+A u=f_\lambda(u)$ on a Banach space $X$, where $A$ is a sectorial operator, and $\lambda\in R$ is the bifurcation parameter.…
The aim of this paper is to study global bifurcations of non-constant solutions of some nonlinear elliptic systems, namely the system on a sphere and the Neumann problem on a ball. We study the bifurcation phenomenon from families of…
In this paper, we consider the following nonlinear Schr\"odinger system in $R^3$: \begin{align*} -\Delta u_j +P_j(x) u=\mu_j u_j^3+\sum\limits_{i=1,i\neq j}^N\beta_{ij}u_i^2u_j, \end{align*} where $N\geq3$, $P_j$ are nonnegative radial…
We consider the Paneitz-type equation $\Delta^2 u -\alpha \Delta u +\beta (u-u^q ) =0$ on a closed Riemannian manifold $(M,g)$. We reduce the equation to a fourth-order ordinary differential equation assuming that $(M,g)$ admits a proper…
We study the nonlinear Schr\"odinger system \[ \begin{cases} \displaystyle iu_t+\Delta u-u+(\frac{1}{9}|u|^2+2|w|^2)u+\frac{1}{3}\overline{u}^2w=0,\\ i\displaystyle \sigma w_t+\Delta w-\mu w+(9|w|^2+2|u|^2)w+\frac{1}{9}u^3=0, \end{cases} \]…
We establish the existence of infinitely many nonnegative, segregated solutions for the sublinearly coupled Schr\"odinger system \begin{equation*} \left\{\begin{aligned}-\Delta u+K_1(x)u&=\mu u^{p-1}+ (\sigma_1+1)\beta…
We consider the stationary solutions for a class of Schroedinger equations with a symmetric double-well potential and a nonlinear perturbation. Here, in the semiclassical limit we prove that the reduction to a finite-mode approximation give…
We consider positive solutions, possibly unbounded, to the semilinear equation $-\Delta u=f(u)$ on continuous epigraphs bounded from below. Under the homogeneous Dirichlet boundary condition, we prove new monotonicity results for $u$, when…
We introduce a model motivated by studies of Bose-Einstein condensates (BECs) trapped in double-well potentials. We assume that a mixture of two hyperfine states of the same atomic species is loaded in such a trap.The analysis is focused on…
In this work we study the following class of systems of coupled nonlinear fractional nonlinear Schr\"odinger equations, \begin{equation*} \left \{ \begin{array}{l} (-\Delta)^s u_1+ \lambda_1 u_1= \mu_1 |u_1|^{2p-2}u_1+\beta |u_2|^{p}…