Tobias König

activity

20172026

most citedClassification of positive solutions to a nonlinear biharmonic equation with critical exponent

50 citations · 119 across the 18 of their papers we have counts for

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18 papers · 1 filter

math.AP2024

Sharp extinction rates for positive solutions of fast diffusion equations

Tobias König, Meng Yu

Let $s \in (0, 1]$ and $N > 2 s$ . It is known that positive solutions to the (fractional) fast diffusion equation $\partial_{t} u + (- Δ)^{s} (u^{\frac{N - 2 s}{N + 2 s}}) = 0$ on $(0, \infty) \…

math.AP2024

Quantitative stability of the total $Q$ -curvature near minimizing metrics

João Henrique Andrade, Tobias König, Jesse Ratzkin +1

Under appropriate positivity hypotheses, we prove quantitative estimates for the total $k$ -th order $Q$ -curvature functional near minimizing metrics on any smooth, closed $n$ -dimen…

math.AP2024

Non-degeneracy, stability and symmetry for the fractional Caffarelli-Kohn-Nirenberg inequality

Nicola De Nitti, Federico Glaudo, Tobias König

The fractional Caffarelli-Kohn-Nirenberg inequality states that $$ \int_{\mathbb{R}^n}\int_{\mathbb{R}^n} \frac{(u(x)-u(y))^2}{|x|^α|x-y|^{n+2s} |y|^α} \mathrm{d} x \, \mathrm{d} y…

math.AP2023★ 5 cited

An exceptional property of the one-dimensional Bianchi-Egnell inequality

Tobias König

In this paper, for $d \geq 1$ and $s \in (0, \frac{d}{2})$ , we study the Bianchi-Egnell quotient \[ \mathcal Q(f) = \inf_{f \in \dot{H}^s(\mathbb R^d) \setminus \mathcal B} \frac{\|…

math.AP2022★ 28 cited

On the sharp constant in the Bianchi-Egnell stability inequality

Tobias König

This note is concerned with the Bianchi-Egnell inequality, which quantifies the stability of the Sobolev inequality, and its generalization to fractional exponents $s \in (0, \frac…

math.AP2022★ 4 cited

Stability for the Sobolev inequality: existence of a minimizer

Tobias König

We prove that the stability inequality associated to Sobolev's inequality and its set of optimizers $M$ and given by \[ \frac{\|\nabla f\|_{L^2(\mathbb R^d)}^2 - S_d \|f\|…

math.AP2022★ 11 cited

Stability with explicit constants of the critical points of the fractional Sobolev inequality and applications to fast diffusion

Nicola De Nitti, Tobias König

We study the quantitative stability of critical points of the fractional Sobolev inequality. We show that, for a non-negative function $u \in H ˙^{s} (R^{N})$ whose energy sa…

math.AP2022★ 10 cited

Fine multibubble analysis in the higher-dimensional Brezis-Nirenberg problem

Tobias König, Paul Laurain

For a bounded set $Ω \subset R^{N}$ and a perturbation $V \in C^{1} (Ω)$ , we analyze the concentration behavior of a blow-up sequence of positive solutions to \[ -Δu_ε+…

math.AP2022

Singular metrics of constant negative $Q$ -curvature in Euclidean spaces

Tobias König, Yamin Wang

We study singular metrics of constant negative $Q$ -curvature in the Euclidean space $R^{n}$ for every $n \geq 1$ . Precisely, we consider solutions to the problem \[ (-Δ)^{n/…

math.AP2022★ 2 cited

Multibubble blow-up analysis for the Brezis-Nirenberg problem in three dimensions

Tobias König, Paul Laurain

For a smooth bounded domain $Ω \subset R^{3}$ and smooth functions $a$ and $V$ , we consider the asymptotic behavior of a sequence of positive solutions $u_{ε}$ to $-Δu_ε+ (a+εV)…

math.AP2021★ 6 cited

Critical functions and blow-up asymptotics for the fractional Brezis--Nirenberg problem in low dimension

Nicola De Nitti, Tobias König

For $s \in (0, 1)$ and a bounded open set $Ω \subset R^{N}$ with $N > 2 s$ , we study the fractional Brezis--Nirenberg type minimization problem of finding $$ S(a) := \inf \frac{…

math.AP2021★ 2 cited

Constant $Q$ -curvature metrics with a singularity

Tobias König, Paul Laurain

For dimensions $n \geq 3$ , we classify singular solutions to the generalized Liouville equation $(- Δ)^{n /2} u = e^{n u}$ on $R^{n} ∖ {0}$ with the finite integral c…

math.AP2021

Reverse conformally invariant Sobolev inequalities on the sphere

Rupert L. Frank, Tobias König, Hanli Tang

We consider the optimization problem corresponding to the sharp constant in a conformally invariant Sobolev inequality on the $n$ -sphere involving an operator of order $2 s > n$ . In…

math.AP2020★ 1 cited

Classification of solutions of an equation related to a conformal log Sobolev inequality

Rupert L. Frank, Tobias König, Hanli Tang

We classify all finite energy solutions of an equation which arises as the Euler--Lagrange equation of a conformally invariant logarithmic Sobolev inequality on the sphere due to B…

math.AP2019

Energy asymptotics in the Brezis-Nirenberg problem. The higher-dimensional case

Rupert Frank, Tobias König, Hynek Kovarik

For dimensions $N \geq 4$ , we consider the Brézis-Nirenberg variational problem of finding \[ S(εV) := \inf_{0\not\equiv u\in H^1_0(Ω)} \frac{\int_Ω|\nabla u|^2 \, dx +ε\int_ΩV\, |…

math.AP2019

Energy asymptotics in the three-dimensional Brezis--Nirenberg problem

Rupert L. Frank, Tobias König, Hynek Kovarik

For a bounded open set $Ω \subset R^{3}$ we consider the minimization problem $$ S(a+εV) = \inf_{0\not\equiv u\in H^1_0(Ω)} \frac{\int_Ω(|\nabla u|^2+ (a+εV) |u|^2)\,dx}{(\int_…

math.AP2019

Singular solutions to a semilinear biharmonic equation with a general critical nonlinearity

Rupert L. Frank, Tobias König

We consider positive solutions $u$ of the semilinear biharmonic equation $Δ^{2} u = ∣ x ∣^{- \frac{n + 4}{2}} g (∣ x ∣^{\frac{n - 4}{2}} u)$ in $R^{n} ∖ {0}$ with non-removable s…

math.AP2017★ 50 cited

Classification of positive solutions to a nonlinear biharmonic equation with critical exponent

Rupert L. Frank, Tobias König

For $n \geq 5$ , we consider positive solutions $u$ of the biharmonic equation \[ Δ^2 u = u^\frac{n+4}{n-4} \qquad \text{on}\ \mathbb R^n \setminus \{0\} \] with a non-removable sin…

Sharp extinction rates for positive solutions of fast diffusion equations

Quantitative stability of the total Q-curvature near minimizing metrics

Non-degeneracy, stability and symmetry for the fractional Caffarelli-Kohn-Nirenberg inequality

An exceptional property of the one-dimensional Bianchi-Egnell inequality

On the sharp constant in the Bianchi-Egnell stability inequality

Stability for the Sobolev inequality: existence of a minimizer

Stability with explicit constants of the critical points of the fractional Sobolev inequality and applications to fast diffusion

Fine multibubble analysis in the higher-dimensional Brezis-Nirenberg problem

Singular metrics of constant negative Q-curvature in Euclidean spaces

Multibubble blow-up analysis for the Brezis-Nirenberg problem in three dimensions

Critical functions and blow-up asymptotics for the fractional Brezis--Nirenberg problem in low dimension

Constant Q-curvature metrics with a singularity

Reverse conformally invariant Sobolev inequalities on the sphere

Classification of solutions of an equation related to a conformal log Sobolev inequality

Energy asymptotics in the Brezis-Nirenberg problem. The higher-dimensional case

Energy asymptotics in the three-dimensional Brezis--Nirenberg problem

Singular solutions to a semilinear biharmonic equation with a general critical nonlinearity

Classification of positive solutions to a nonlinear biharmonic equation with critical exponent

Quantitative stability of the total $Q$ -curvature near minimizing metrics

Singular metrics of constant negative $Q$ -curvature in Euclidean spaces

Constant $Q$ -curvature metrics with a singularity