Searcharxiv⌕ Search

arXiv subjects

Tobias Weth

Publications and source records attributed to Tobias Weth.

At least 55 records · Page 3Linked to original sources
arXiv2020

Spiraling solutions of nonlinear Schrödinger equations

We study a new family of sign-changing solutions to the stationary nonlinear Schrödinger equation $$ -Δv +q v =|v|^{p-2} v, \qquad \text{in $\mathbb{R}^3$,} $$ with $2<p<\infty$ and $q \ge 0$. These solutions are spiraling in the sense that they are not axially symmetric but invariant under screw motion, i.e., they share the symmetry properties of a helicoid. In addition to existence results, we provide information on the shape of spiraling solutions, which depends on the parameter value representing the rotational slope of the underlying screw motion. Our results complement a related analysis of Del Pino, Musso and Pacard for the Allen-Cahn equation, whereas the nature of results and the underlying variational structure are completely different.

math.AP↗
arXiv2020

Nodal Solutions for sublinear-type problems with Dirichlet boundary conditions

We consider nonlinear second order elliptic problems of the type \[ -Δu=f(u) \text{ in } Ω, \qquad u=0 \text{ on } \partial Ω, \] where $Ω$ is an open $C^{1,1}$-domain in $\mathbb{R}^N$, $N\geq 2$, under some general assumptions on the nonlinearity that include the case of a sublinear pure power $f(s)=|s|^{p-1}s$ with $0 1$ and $λ>λ_2(Ω)$ (the second Dirichlet eigenvalue of the Laplacian). We prove the existence of a least energy nodal (i.e. sign changing) solution, and of a nodal solution of mountain-pass type. We then give explicit examples of domains where the associated levels do not coincide. For the case where $Ω$ is a ball or annulus and $f$ is of class $C^1$, we prove instead that the levels coincide, and that least energy nodal solutions are nonradial but axially symmetric functions. Finally, we provide stronger results for the Allen-Cahn type nonlinearities in case $Ω$ is either a ball or a square. In particular we give a complete description of the solution set for $λ\sim λ_2(Ω)$, computing the Morse index of the solutions.

math.AP↗
arXiv2019

The Dirichlet Problem for the Logarithmic Laplacian

In this paper, we study the logarithmic Laplacian operator $L_Δ$, which is a singular integral operator with symbol $2\log |ζ|$. We show that this operator has the integral representation $$L_Δu(x) = c_{N} \int_{\mathbb{R}^N } \frac{ u(x)1_{B_1(x)}(y)-u(y)}{|x-y|^{N} } dy + ρ_N u(x) $$ with $c_N = π^{- \frac{N}{2}} Γ(\frac{N}{2})$ and $ρ_N=2 \log 2 + ψ(\frac{N}{2}) -γ$, where $Γ$ is the Gamma function, $ψ= \frac{Γ'}Γ$ is the Digamma function and $γ= -Γ'(1)$ is the Euler Mascheroni constant. This operator arises as formal derivative $\partial_s \Big|_{s=0} (-Δ)^s$ of fractional Laplacians at $s= 0$. We develop the functional analytic framework for Dirichlet problems involving the logarithmic Laplacian on bounded domains and use it to characterize the asymptotics of principal Dirichlet eigenvalues and eigenfunctions of $(-Δ)^s$ as $s \to 0$. As a byproduct, we then derive a Faber-Krahn type inequality for the principal Dirichlet eigenvalue of $L_Δ$. Using this inequality, we also establish conditions on domains giving rise to the maximum principle in weak and strong forms. This allows us to also derive regularity up to the boundary of solutions to corresponding Poisson problems.

math.AP↗
arXiv2019

Spectral asymptotics of radial solutions and nonradial bifurcation for the Hénon equation

We study the spectral asymptotics of nodal (i.e., sign-changing) solutions of the problem \begin{equation*} (H) \qquad \qquad \left \{ \begin{aligned} -Δu &=|x|^α|u|^{p-2}u&&\qquad \text{in ${\bf B}$,} \\ u&=0&&\qquad \text{on $\partial {\bf B}$,} \end{aligned} \right. \end{equation*} in the unit ball ${\bf B} \subset \mathbb{R}^N,N\geq 3$, $p>2$ in the limit $α\to +\infty$. More precisely, for a given positive integer $K$, we derive asymptotic $C^1$-expansions for the negative eigenvalues of the linearization of the unique radial solution $u_α$ of $(H)$ with precisely $K$ nodal domains and $u_α(0)>0$. As an application, we derive the existence of an unbounded sequence of bifurcation points on the radial solution branch $α\mapsto (α,u_α)$ which all give rise to bifurcation of nonradial solutions whose nodal sets remain homeomorphic to a disjoint union of concentric spheres.

math.AP↗
arXiv2018

Local compactness and nonvanishing for weakly singular nonlocal quadratic forms

In this work we study a class of nonlocal quadratic forms given by \[ \mathcal{E}_j(u,v)=\frac{1}{2}\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}(u(x)-u(y))(v(x)-v(y))j(x-y)\ dxdy, \] where $j:\mathbb{R}^N\to[0,\infty]$ is a measurable even function with $\min\{1,|\cdot|^2\}j\in L^1(\mathbb{R}^N)$. Assuming merely $j\notin L^1(\mathbb{R}^N)$, we show local compactness of the embedding $\mathcal{D}^j(\mathbb{R}^N)\hookrightarrow L^2(\mathbb{R}^N)$, where $\mathcal{D}^j(\mathbb{R}^N)$ denotes the space of functions $u\in L^2(\mathbb{R}^N)$ with $\mathcal{E}_j(u,u)<\infty$. Using this local compactness, we establish an alternative which allows to distinguish vanishing and nonvanishing of bounded sequences in $\mathcal{D}^j(\mathbb{R}^N)$. As an application, we show the existence of maximizers for a class of integral functionals defined on the unit sphere in $\mathcal{D}^j(\mathbb{R}^N)$. Our main results extend to cylindrical unbounded sets of the type $Ω= U \times \mathbb{R}^k$, where $U \subset \mathbb{R}^{N-k}$ is open and bounded. Finally, we note that a Poincaré inequality associated with $\mathcal{E}_j$ holds for unbounded domains of this type, thereby extending previously known results for bounded domains.

math.AP↗
arXiv2018

On the strong maximum principle for nonlocal operators

In this paper we derive a strong maximum principle for weak supersolutions of nonlocal equations of the form $Iu=c(x) u$ in $Ω$, where $Ω\subset \mathbb{R}^N$ is a domain, $c\in L^{\infty}(Ω)$ and $I$ is an operator of the form $Iu(x)=P.V.\int_{\mathbb{R}^N}(u(x)-u(y))j(x-y)\ dy$ with a nonnegative kernel function $j$. We formulate minimal positivity assumptions on $j$ corresponding to a class of operators which includes highly anisotropic variants of the fractional Laplacian. Somewhat surprisingly, this problem leads to the study of general lattices in $\mathbb{R}^N$. Our results extend to the regional variant of the operator $I$ and, under weak additional assumptions, also to the case of $x$-dependent kernel functions.

math.AP↗
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↗
arXiv2018

Critical domains for the first nonzero Neumann eigenvalue in Riemannian manifolds

The present paper is devoted to geometric optimization problems related to the Neumann eigenvalue problem for the Laplace-Beltrami operator on bounded subdomains $Ω$ of a Riemannian manifold $(\mathcal{M},g)$. More precisely, we analyze locally extremal domains for the first nontrivial eigenvalue $μ_2(Ω)$ with respect to volume preserving domain perturbations, and we show that corresponding notions of criticality arise in the form of overdetermined boundary problems. Our results rely on an extension of Zanger's shape derivative formula which covers the case when $μ_2(Ω)$ is not a simple eigenvalue. In the second part of the paper, we focus on product manifolds of the form $\mathcal{M} = \mathbb{R}^k \times \mathcal{N}$, and we classify the subdomains where an associated overdetermined boundary value problem has a solution.

math.AP↗
arXiv2018

Symmetry breaking via Morse index for equations and systems of Hénon-Schrödinger type

We consider the Dirichlet problem for the Schrödinger-Hénon system $$ -Δu + μ_1 u = |x|^α\partial_u F(u,v),\quad \qquad -Δv + μ_2 v = |x|^α\partial_v F(u,v) $$ in the unit ball $Ω\subset \mathbb{R}^N, N\geq 2$, where $α>-1$ is a parameter and $F: \mathbb{R}^2 \to \mathbb{R}$ is a $p$-homogeneous $C^2$-function for some $p>2$ with $F(u,v)>0$ for $(u,v) \not = (0,0)$. We show that, as $α\to \infty$, the Morse index of nontrivial radial solutions of this problem (positive or sign-changing) tends to infinity. This result is new even for the corresponding scalar Hénon equation and extends a previous result by Moreira dos Santos and Pacella for the case $N=2$. In particular, the result implies symmetry breaking for ground state solutions, but also for other solutions obtained by an $α$-independent variational minimax principle.

math.AP↗
arXiv2017

Serrin's overdetermined problem on the sphere

We study Serrin's overdetermined boundary value problem \begin{equation*} -Δ_{S^N}\, u=1 \quad \text{ in $Ω$},\qquad u=0, \; \partial_ηu=\textrm{const} \quad \text{on $\partial Ω$} \end{equation*} in subdomains $Ω$ of the round unit sphere $S^N \subset \mathbb{R}^{N+1}$, where $Δ_{S^N}$ denotes the Laplace-Beltrami operator on $S^N$. A subdomain $Ω$ of $S^N$ is called a Serrin domain if it admits a solution of this overdetermined problem. In our main result, we construct Serrin domains in $S^N$, $N \ge 2$ which bifurcate from symmetric straight tubular neighborhoods of the equator. Our result provides the first example of Serrin domains in $S^{N}$ which are not bounded by geodesic spheres.

math.AP↗
arXiv2017

Liouville theorems for a general class of nonlocal operators

In this paper, we study the equation $\mathcal{L} u=0$ in $\mathbb{R}^N$, where $\mathcal{L}$ belongs to a general class of nonlocal linear operators which may be anisotropic and nonsymmetric. We classify distributional solutions of this equation, thereby extending and generalizing recent Liouville type theorems in the case where $\mathcal{L}= (-Δ)^s$, $s \in (0,1)$ is the classical fractional Laplacian.

math.AP↗
arXiv2017

The unique continuation property of sublinear equations

We derive the unique continuation property of a class of semi-linear elliptic equations with non-Lipschitz nonlinearities. The simplest type of equations to which our results apply is given as $-Δu = |u|^{σ-1} u$ in a domain $Ω\subset \mathbb{R}^N$, with $0 \le σ<1$. Despite the sublinear character of the nonlinear term, we prove that if a solution vanishes in an open subset of $Ω$, then it vanishes necessarily in the whole $Ω$. We then extend the result to equations with variable coefficients operators and inhomogeneous right-hand side.

math.AP↗
arXiv2017

Delaunay hypersurfaces with constant nonlocal mean curvature

We study hypersurfaces of $\mathbb{R}^N$ with constant nonlocal (or fractional) mean curvature. This is the equation associated to critical points of the fractional perimeter functional under a volume constraint. We establish the existence of a smooth branch of periodic cylinders in $\mathbb{R}^N$, $N\geq 2$, all of them with the same constant nonlocal mean curvature, and bifurcating from a straight cylinder. These are Delaunay type cylinders in the nonlocal setting. The proof uses the Crandall-Rabinowitz theorem applied to a quasilinear type fractional elliptic equation.

math.DG↗
arXiv2017

Ground states and high energy solutions of the planar Schrödinger-Poisson system

In this paper, we are concerned with the Schrödinger-Poisson system \begin{equation} (0.1)\qquad -Δu + u +ϕu = |u|^{p-2}u \quad \text{in}\ \mathbb{R}^{d},\qquad Δϕ= u^{2} \quad \text{in}\ \mathbb{R}^{d}. \end{equation} Due to its relevance in physics, the system has been extensively studied and is quite well understood in the case $d \ge 3$. In contrast, much less information is available in the planar case $d=2$ which is the focus of the present paper. It has been observed by Cingolani and the second author \cite{Cingolani-Weth-2016} that the variational structure of $(0.1)$ differs substantially in the case $d=2$ and leads to a richer structure of the set of solutions. However, the variational approach of \cite{Cingolani-Weth-2016} is restricted to the case $p \ge 4$ which excludes some physically relevant exponents. In the present paper, we remove this unpleasant restriction and explore the more complicated underlying functional geometry in the case $2<p<4$ with a different variational approach.

math.AP↗
arXiv2017

Near-sphere lattices with constant nonlocal mean curvature

We are concerned with unbounded sets of $\mathbb{R}^N$ whose boundary has constant nonlocal (or fractional) mean curvature, which we call CNMC sets. This is the equation associated to critical points of the fractional perimeter functional under a volume constraint. We construct CNMC sets which are the countable union of a certain bounded domain and all its translations through a periodic integer lattice of dimension $M\leq N$. Our CNMC sets form a $C^2$ branch emanating from the unit ball alone and where the parameter in the branch is essentially the distance to the closest lattice point. Thus, the new translated near-balls (or near-spheres) appear from infinity. We find their exact asymptotic shape as the parameter tends to infinity.

math.AP↗
arXiv2016

Branch continuation inside the essential spectrum for the nonlinear Schrödinger equation

We consider the nonlinear stationary Schrödinger equation \begin{equation*} -Δu -λu= Q(x)|u|^{p-2}u, \qquad \text{in }\mathbb{R}^N \end{equation*} in the case where $N \geq 3$, $p$ is a superlinear, subcritical exponent, $Q$ is a bounded, nonnegative and nontrivial weight function with compact support in $\mathbb{R}^N$ and $λ\in \mathbb{R}$ is a parameter. Under further restrictions either on the exponent $p$ or on the shape of $Q$, we establish the existence of a continuous branch $\mathcal{C}$ of nontrivial solutions to this equation which intersects $\{λ\} \times L^{s}(\mathbb{R}^N)$ for every $λ\in (-\infty, λ_Q)$ and $s> \frac{2N}{N-1}$. Here $λ_Q>0$ is an explicit positive constant which only depends on $N$ and $\text{diam}(\text{supp }Q)$. In particular, the set of values $λ$ along the branch enters the essential spectrum of the operator $-Δ$.

math.AP↗
arXiv2016

Unbounded periodic solutions to Serrin's overdetermined boundary value problem

We study the existence of nontrivial unbounded domains $Ω$ in $\mathbb{R}^N$ such that the overdetermined problem $$ -Δu = 1 \quad \text{in $Ω$}, \qquad u=0, \quad \partial_νu=\textrm{const} \qquad \text{on $\partial Ω$} $$ admits a solution $u$. By this, we complement Serrin's classification result from 1971 which yields that every bounded domain admitting a solution of the above problem is a ball in $\mathbb{R}^N$. The domains we construct are periodic in some variables and radial in the other variables, and they bifurcate from a straight (generalized) cylinder or slab. We also show that these domains are uniquely self Cheeger relative to a period cell for the problem.

math.AP↗