arXiv subjects

Jarosław Mederski

Publications and source records attributed to Jarosław Mederski.

39 records · Page 3Linked to original sources

arXiv2015

Ground states of nonlinear Schrödinger equations with sum of periodic and inverse-square potentials

We study the existence of solutions of the following nonlinear Schrödinger equation \begin{equation*} -Δu + \Big(V(x)-\fracμ{|x|^2}\Big) u = f(x,u) \hbox{ for } x\in\mathbb{R}^N\setminus\{0\}, \end{equation*} where $V:\mathbb{R}^N\to\mathbb{R}$ and $f:\mathrm{R}^N\times\mathbb{R}\to\mathbb{R}$ are periodic in $x\in\mathbb{R}$. We assume that $0$ does not lie in the spectrum of $-Δ+V$ and $μ<\frac{(N-2)^2}{4}$, $N\geq 3$. The superlinear and subcritical term $f$ satisfies a weak monotonicity condition. For sufficiently small $μ\geq 0$ we find a ground state solution as a minimizer of the energy functional on a natural constraint. If $μ<0$ and $0$ lies below the spectrum of $-Δ+V$, then ground state solutions do not exist.

math.AP↗

arXiv2014

Positive ground states for a system of Schrödinger equations with critically growing nonlinearities

We study the following problem \[ \begin{cases} -Δu = λu + u^{2^*-2} v & \hbox{in} Ω,\\ -Δv= μv^{2^*-1} + u^{2^*-1} & \hbox{in} Ω,\\ u> 0,v> 0 & \hbox{in} Ω,\\ u=v=0 & \hbox{on} \partial Ω, \end{cases} \] where $Ω$ is a bounded domain of $\mathbb{R}^N$, $N\geq 4$, $2^*=2N/(N-2)$, $λ\in\mathbb{R}$ and $μ\geq 0$ and we obtain existence and nonexistence results, depending on the value of the parameters $λ$ and $μ$.

math.AP↗

arXiv2014

Solutions to a nonlinear Schrödinger equation with periodic potential and zero on the boundary of the spectrum

We study the following nonlinear Schrödinger equation $$-Δu + V(x) u = g(x,u),$$ where V and g are periodic in x. We assume that 0 is a right boundary point of the essential spectrum of $-Δ+V$. The superlinear and subcritical term g satisfies a Nehari type monotonicity condition. We employ a Nehari manifold type technique in a strongly indefitnite setting and obtain the existence of a ground state solution. Moreover we get infinitely many geometrically distinct solutions provided that g is odd.

math.AP↗