Related papers: Dynamical and variational properties of the NLS-$\…
We study standing waves for a nonlinear Schr\"odinger equation on a star graph {$\mathcal{G}$} i.e. $N$ half-lines joined at a vertex. At the vertex an interaction occurs described by a boundary condition of delta type with strength…
We study the nonlinear Schr\"odinger equation (NLS) on a star graph $\mathcal{G}$. At the vertex an interaction occurs described by a boundary condition of delta type with strength $\alpha\in \mathbb{R}$. We investigate an orbital…
We study a nonlinear Schr\"odinger equation with logarithmic nonlinearity on a star graph $\mathcal{G}$. At the vertex an interaction occurs described by a boundary condition of delta type with strength $\alpha\in \mathbb{R}$. We…
We investigate the existence of ground states for the nonlinear Schr\"odinger Equation on star graphs with two subcritical focusing nonlinear terms: a standard power nonlinearity, and a delta-type nonlinearity located at the vertex. We find…
We consider variational and stability properties of a system of two coupled nonlinear Schr\"{o}dinger equations on the star graph $\Gamma$ with the $\delta$ coupling at the vertex of $\Gamma$. The first part is devoted to the proof of an…
We study the nonlinear Schr\"odinger equation with an arbitrary real potential $V(x)\in (L^1+L^\infty)(\Gamma)$ on a star graph $\Gamma$. At the vertex an interaction occurs described by the generalized Kirchhoff condition with strength…
We study the orbital stability of action ground-states of the nonlinear Schr\"odinger equation over two particular cases of metric graphs, the $\mathcal{T}$ and the tadpole graphs. We show the existence of stability transitions near the…
We determine and study the ground states of a focusing Schr\"odinger equation in dimension one with a power nonlinearity $|\psi|^{2\mu} \psi$ and a strong inhomogeneity represented by a singular point perturbation, the so-called…
We study strong instability (by blow-up) of the standing waves for the nonlinear Schr\"odinger equation with $\delta$-interaction on a star graph $\Gamma$. The key ingredient is a novel variational technique applied to the standing wave…
In this work, we investigate the existence and orbital (in)stability of several branches of standing--wave solutions for the cubic nonlinear Schr\"odinger equation (NLS) posed on a looping--edge graph $\mathcal{G}$, consisting of a circle…
We consider a half-soliton stationary state of the nonlinear Schrodinger equation with the power nonlinearity on a star graph consisting of N edges and a single vertex. For the subcritical power nonlinearity, the half-soliton state is a…
We consider the nonlinear Schr\"odinger equation with pure power nonlinearity on a general compact metric graph, and in particular its stationary solutions with fixed mass. Since the graph is compact, for every value of the mass there is a…
On a star graph made of $N \geq 3$ halflines (edges) we consider a Schr\"odinger equation with a subcritical power-type nonlinearity and an attractive delta interaction located at the vertex. From previous works it is known that there…
We consider a nonlinear Schr\"odinger equation (NLS) posed on a graph or network composed of a generic compact part to which a finite number of half-lines are attached. We call this structure a starlike graph. At the vertices of the graph…
We study the focusing inhomogeneous nonlinear Schr\"odinger equation $$ i\partial_t u + \Delta u = -|x|^b |u|^{p-1}u ,\quad (t,x)\in (0,\infty)\times\mathbb{R}^N, $$ with $b>0$ and $p>1$. Due to the spatial growth of the nonlinearity,…
We study existence and properties of ground states for the nonlinear Schr\"odinger equation with combined power nonlinearities \[ -\Delta u= \lambda u + \mu |u|^{q-2} u + |u|^{p-2} u \qquad \text{in $\mathbb{R}^N$, $N \ge 1$,} \] having…
In this paper, we study the ground state solutions of the following coupled nonlinear Schr\"odinger system (P) $-\Delta u_1-\tau_1 u_1 =\mu_1u_1^3+\beta u_1u_2^2$, $ -\Delta u_2-\tau_2 u_2 =\mu_2u_2^3+\beta u_1^2u_2$ in $\Omega$,…
We consider a nonlinear Schr\"odinger equation with focusing nonlinearity of power type on a star graph ${\mathcal G}$, written as $ i \partial_t \Psi (t) = H \Psi (t) - |\Psi (t)|^{2\mu}\Psi (t)$, where $H$ is the selfadjoint operator…
In this paper we deal with the following weakly coupled nonlinear Schr\"{o}dinger system \begin{align*} \begin{cases} - \Delta_\alpha u + \omega u = |u|^2 u + \beta u |v|^2&\quad \mathrm{in}\ \mathbb{R}^2,\\ - \Delta v + \tilde{\omega} v =…
We investigate the ground states of the one-dimensional nonlinear Schr\"odinger equation with a defect located at a fixed point. The nonlinearity is focusing and consists of a subcritical power. The notion of ground state can be defined in…