arXiv subjects

Tobias König

Publications and source records attributed to Tobias König.

23 records · Page 2Linked to original sources

arXiv2019

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

For dimensions $N \geq 4$, we consider the Brézis-Nirenberg variational problem of finding \[ S(εV) := \inf_{0\not\equiv u\in H^1_0(Ω)} \frac{\int_Ω|\nabla u|^2 \, dx +ε\int_ΩV\, |u|^2 \, dx}{\left(\int_Ω|u|^q \, dx \right)^{2/q}}, \] where $q=\frac{2N}{N-2}$ is the critical Sobolev exponent and $Ω\subset \mathbb{R}^N$ is a bounded open set. We compute the asymptotics of $S(0) - S(εV)$ to leading order as $ε\to 0+$. We give a precise description of the blow-up profile of (almost) minimizing sequences and, in particular, we characterize the concentration points as being extrema of a quotient involving the Robin function. This complements the results from our recent paper in the case $N = 3$.

math.AP↗

arXiv2019

Liquid Drop Model for Nuclear Matter in the Dilute Limit

We consider the liquid drop model for nuclei interacting with a neutralizing homogeneous background of electrons. The regime we are interested in is when the fraction between the electronic and the nuclear charge density is small. We show that in this dilute limit the thermodynamic ground state energy is given to leading order by that of an isolated nucleus.

math-ph↗

arXiv2019

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

For a bounded open set $Ω\subset\mathbb R^3$ we consider the minimization problem $$ S(a+εV) = \inf_{0\not\equiv u\in H^1_0(Ω)} \frac{\int_Ω(|\nabla u|^2+ (a+εV) |u|^2)\,dx}{(\int_Ωu^6\,dx)^{1/3}} $$ involving the critical Sobolev exponent. The function $a$ is assumed to be critical in the sense of Hebey and Vaugon. Under certain assumptions on $a$ and $V$ we compute the asymptotics of $S(a+εV)-S$ as $ε\to 0+$, where $S$ is the Sobolev constant. (Almost) minimizers concentrate at a point in the zero set of the Robin function corresponding to $a$ and we determine the location of the concentration point within that set. We also show that our assumptions are almost necessary to have $S(a+εV) 0$.

math.AP↗

arXiv2019

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

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 $\mathbb R^n \setminus \{0\}$ with non-removable singularities at the origin. Under natural assumptions on the nonlinearity $g$, we show that $|x|^\frac{n-4}{2} u$ is a periodic function of $\ln |x|$ and we classify all such solutions.

math.AP↗

arXiv2017

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

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 singularity at the origin. We show that $|x|^{\frac{n-4}{2}} u$ is a periodic function of $\ln |x|$ and we classify all periodic functions obtained in this way. This result is relevant for the description of the asymptotic behavior near singularities and for the $Q$-curvature problem in conformal geometry.

math.AP↗