Searcharxiv⌕ Search

arXiv subjects

Nils Ackermann

Publications and source records attributed to Nils Ackermann.

6 recordsLinked to original sources
arXiv2018

Unstable normalized standing waves for the space periodic NLS

For the stationary nonlinear Schrödinger equation $-Δu+ V(x)u- f(u) = λu$ with periodic potential $V$ we study the existence and stability properties of multibump solutions with prescribed $L^2$-norm. To this end we introduce a new nondegeneracy condition and develop new superposition techniques which allow to match the $L^2$-constraint. In this way we obtain the existence of infinitely many geometrically distinct solutions to the stationary problem. We then calculate the Morse index of these solutions with respect to the restriction of the underlying energy functional to the associated $L^2$-sphere, and we show their orbital instability with respect to the Schrödinger flow. Our results apply in both, the mass-subcritical and the mass-supercritical regime.

math.AP↗
arXiv2015

Precise exponential decay for solutions of semilinear elliptic equations and its effect on the structure of the solution set for a real analytic nonlinearity

We are concerned with the properties of weak solutions of the stationary Schrödinger equation $-Δu + Vu = f(u)$, $u\in H^1(\mathbb{R}^N)\cap L^\infty(\mathbb{R}^N)$, where $V$ is Hölder continuous and $\inf V>0$. Assuming $f$ to be continuous and bounded near $0$ by a power function with exponent larger than $1$ we provide precise decay estimates at infinity for solutions in terms of Green's function of the Schrödinger operator. In some cases this improves known theorems on the decay of solutions. If $f$ is also real analytic on $(0,\infty)$ we obtain that the set of positive solutions is locally path connected. For a periodic potential $V$ this implies that the standard variational functional has discrete critical values in the low energy range and that a compact isolated set of positive solutions exists, under additional assumptions.

math.AP↗
arXiv2012

Boundary clustered layers near the higher critical exponents

We consider the supercritical problem {equation*} -Δu=|u| ^{p-2}u\text{\in}Ω,\quad u=0\text{\on}\partialΩ, {equation*} where $Ω$ is a bounded smooth domain in $\mathbb{R}^{N}$ and $p$ smaller than the critical exponent $2_{N,k}^{\ast}:=\frac{2(N-k)}{N-k-2}$ for the Sobolev embedding of $H^{1}(\mathbb{R}^{N-k})$ in $L^{q}(\mathbb{R}^{N-k})$, $1\leq k\leq N-3.$ We show that in some suitable domains $Ω$ there are positive and sign changing solutions with positive and negative layers which concentrate along one or several $k$-dimensional submanifolds of $\partialΩ$ as $p$ approaches $2_{N,k}^{\ast}$ from below. Key words:Nonlinear elliptic boundary value problem; critical and supercritical exponents; existence of positive and sign changing solutions.

math.AP↗
arXiv2012

Alternating sign multibump solutions of nonlinear elliptic equations in expanding tubular domains

Let $Γ$ denote a smooth simple curve in $\mathbb{R}^{N}$, $N\geq2$, possibly with boundary. Let $Ω_{R}$ be the open normal tubular neighborhood of radius 1 of the expanded curve $RΓ:=\{Rx\mid x\in Γ\smallsetminus\partialΓ\}$. Consider the superlinear problem $-Δu+λu=f(u)$ on the domains $Ω_{R}$, as $R\rightarrow \infty$, with homogeneous Dirichlet boundary condition. We prove the existence of multibump solutions with bumps lined up along $RΓ$ with alternating signs. The function $f$ is superlinear at 0 and at $\infty$, but it is not assumed to be odd. If the boundary of the curve is nonempty our results give examples of contractible domains in which the problem has multiple sign changing solutions.

math.AP↗
arXiv2012

A concentration phenomenon for semilinear elliptic equations

For a domain $Ω\subset\dR^N$ we consider the equation $ -Δu + V(x)u = Q_n(x)\abs{u}^{p-2}u$ with zero Dirichlet boundary conditions and $p\in(2,2^*)$. Here $V\ge 0$ and $Q_n$ are bounded functions that are positive in a region contained in $Ω$ and negative outside, and such that the sets $\{Q_n>0\}$ shrink to a point $x_0\inΩ$ as $n\to\infty$. We show that if $u_n$ is a nontrivial solution corresponding to $Q_n$, then the sequence $(u_n)$ concentrates at $x_0$ with respect to the $H^1$ and certain $L^q$-norms. We also show that if the sets $\{Q_n>0\}$ shrink to two points and $u_n$ are ground state solutions, then they concentrate at one of these points.

math.AP↗