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Bernhard Ruf

Publications and source records attributed to Bernhard Ruf.

14 recordsLinked to original sources
arXiv2025

Non-uniqueness of mild solutions for 2d-heat equations with singular initial data

In a recent article by the authors [15] it was shown that wide classes of semilinear elliptic equations with exponential type nonlinearities admit singular radial solutions $U$ on the punctured disc in $\mathbb R^2$ which are also distributional solutions on the whole disc. We show here that these solutions, taken as initial data of the associated heat equation, give rise to non-uniqueness of mild solutions: ${u_s}(t,x) \equiv U(x)$ is a stationary solution, and there exists also a solution ${u_r}(t,x)$ departing from $U$ which is bounded for $t > 0$. While such non-uniqueness results have been known in higher dimensions by Ni--Sacks [33], Terraneo [40] and Galaktionov--Vazquez [16], only two very specific results have recently been obtained in two dimensions by Ioku--Ruf--Terraneo [22] and Ibrahim--Kikuchi--Nakanishi--Wei [21].

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arXiv2025

On inequalities of Bliss-Moser type with loss of compactness in $\mathbb{R}^N$

We prove the following Limiting Bliss inequalities \begin{equation}\nonumber \sup\limits_{v(0) = 0, \int_0^1|v'|^Ndx=1 }\int_0^1 e^{\beta\left(\log\frac{e}{s}\right)\frac{v^N(s)}{s^{N-1}}}ds\leq C(N,\beta), \ \hbox{ for } \beta \le 1 \end{equation} The inequalities are optimal with respect to $\beta \le 1$; there is compactness for $\beta<1$, and along the infinitesimal Moser sequence for $\beta = 1$. Moreover, we show that the improved inequalities \begin{equation}\nonumber \sup\limits_{v(0) = 0, \int_0^1|v'|^Ndx=1 }\int_0^1 e^{\left(\log\frac{e}{s}+\gamma\log\log\frac{e}{s}\right)\frac{v^N(s)}{s^{N-1}}}ds\leq C(N,\gamma) \end{equation} hold for $\gamma\leq1$, and for $\gamma=1$ the inequalities are critical with loss of compactness. The inequalities are optimal: no further improvement in the coefficient of the exponent is possible. The second result extends the result in [J. M. do \'{O}, B. Ruf and P. Ubilla, A critical Moser type inequality with loss of compactness due to infinitesimal shocks, Calc. Var. Partial Differential Equations 62 (2023)] from $N=2$ to general dimensions $N\geq2$.

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arXiv2024

Multiplicity results for a subcritical Hamiltonian system with concave-convex nonlinearities

We study the {\it Hamiltonian elliptic system} \begin{eqnarray}\label{HS1-abstract} \left\{ \begin{aligned} -\Delta u & = \lambda |v|^{r-1}v +|v|^{p-1}v \qquad &\hbox{in} \ \ \Omega ,\\ -\Delta v & = \mu |u|^{s-1}u +|u|^{q-1}u \qquad &\hbox{in} \ \ \Omega ,\\ u &>0, \ v>0 \qquad \, &\hbox{in} \ \ \Omega ,\\ u &=v = 0 \qquad \quad &\hbox{on} \quad \partial \Omega, \end{aligned} \right. \end{eqnarray} where $\Omega \subset \mathbb {R}^N$ is a smooth bounded domain, $\lambda$ and $ \mu $ are nonnegative parameters and $r,s,p,q>0$. Our study includes the case in which the nonlinearities in \eqref{HS1-abstract} are concave near the origin and convex near infinity, and we focus on the region of non-negative {\it pairs of parameters} \red{$(\lambda,\mu)$} that guarantee exis\-tence and multiplicity of solutions of \eqref{HS1-abstract}. \red{In particular, we show the existence of a strictly decreasing curve $\lambda_*(\mu)$ on an interval $[0, \mu]$ with $\lambda_*(0)> 0, \lambda_*(\mu) = 0$ and such that the system has two solutions for $(\lambda,\mu)$ below the curve, one solution for $(\lambda, \mu)$ on the curve and no solution for $(\lambda, \mu)$ above the curve. A similar statement holds reversing $\lambda$ and $\mu$.} This work is motivated by some of the results by Ambrosseti, BRezis and Cerami from 1993.

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arXiv2024

Qualitative Properties of Solutions of Semilinear Elliptic Systems

The article explores the qualitative properties of solutions to elliptic equations and systems, focusing particularly on whether solutions retain the symmetry of their domains. According to the well-known Gidas-Ni-Nirenberg theorem, positive solutions to certain autonomous elliptic equations in radial domains are radial themselves. However, this symmetry can be broken in equations with power weight terms. The article also examines related results for systems of these weighted equations.

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arXiv2024

Singular solutions of semilinear elliptic equations with exponential nonlinearities in 2-dimensions

By introducing a new classification of the growth rate of exponential functions, singular solutions for semilinear elliptic equations in 2-dimensions with exponential nonlinearities are constructed. The strategy is to introduce a model nonlinearity which admits an explicit singular solution. Then, using a transformation as in [8], one obtains an approximate singular solution, and then one concludes by a suitable fixed point argument. Our method covers a wide class of nonlinearities in a unified way. As a special case, our result contains a pioneering contribution by Ibrahim--Kikuchi--Nakanishi--Wei [15] for the Moser--Trudinger type nonlinearity.

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arXiv2023

The Spectrum Zero Problem of nonlinear Dirac equation with particle-antiparticle interaction

In this study, we investigate the Spectrum Zero Problem of nonlinear Dirac equations with a focus on the behavior of zero at the boundaries of the spectral gap. We introduce a nonlinear particle-antiparticle interaction and demonstrate that the problem exhibits asymmetric behavior at the left and right boundaries of the spectrum. Specifically, when zero is at the right boundary, the problem has only trivial solutions and is identified as a bifurcation point on the left, whereas nontrivial solutions exist when zero is at the left boundary or within the spectral gap. The main idea is to employ a variational method involving a perturbation technique that places zero within the spectral gap. We use the critical point theorem of the perturbed functional to construct a Palais-Smale sequence in order to approach the critical point of the target energy functional. Additionally, we utilize the concentration-compactness principle to identify critical points of the original functional and explore the associated bifurcation phenomena. Our results reveal an asymmetric phenomenon in nonlinear quantum systems and provide insights into why strongly indefinite problems typically address zero only at the left boundary of the spectral gap.

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arXiv2022

Bifurcation results for nonlinear eigenvalue problems involving the (p,q)-Laplace operator

In this paper, we analyze an eigenvalue problem for nonlinear elliptic operators involving homogeneous Dirichlet boundary conditions in a open smooth bounded domain. We prove bifurcation results from trivial solutions and from infinity for the considered nonlinear eigenvalue problem. We also show the existence of multiple solutions of the nonlinear problem using variational methods.

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arXiv2022

Bifurcation into spectral gaps for strongly indefinite Choquard equations

We consider the semilinear elliptic equations $$ \left\{ \begin{array}{ll} &-Δu+V(x)u=\left(I_α\ast |u|^p\right)|u|^{p-2}u+λu\quad \hbox{for } x\in\mathbb R^N, \\ &u(x) \to 0 \hbox{ as } |x| \to\infty, \end{array} \right. $$ where $I_α$ is a Riesz potential, $p\in(\frac{N+α}N,\frac{N+α}{N-2})$, $N\geq3$, and $V $ is continuous periodic. We assume that $0$ lies in the spectral gap $(a,b)$ of $-Δ+ V$. We prove the existence of infinitely many geometrically distinct solutions in $H^1(\mathbb R^N)$ for each $λ\in(a, b)$, which bifurcate from $b$ if $\frac{N+α}N< p < 1 +\frac{2+α}{N}$. Moreover, $b$ is the unique gap-bifurcation point (from zero) in $[a,b]$. When $λ=a$, we find infinitely many geometrically distinct solutions in $H^2_{loc}(\mathbb R^N)$. Final remarks are given about the eventual occurrence of a bifurcation from infinity in $λ=a$.

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arXiv2021

Nonlinear eigenvalue problems and bifurcation for quasi-linear elliptic operators

In this paper, we analyze an eigenvalue problem for quasi-linear elliptic operators involving homogeneous Dirichlet boundary conditions in a open smooth bounded domain. We show that the eigenfunctions corresponding to the eigenvalues belong to $L^{\infty}$, which implies $C^{1,α}$ smoothness, and the first eigenvalue is simple. Moreover, we investigate the bifurcation results from trivial solutions using the Krasnoselski bifurcation theorem and from infinity using the Leray-Schauder degree. We also show the existence of multiple critical points using variational methods and the Krasnoselski genus.

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arXiv2019

Non-uniqueness for a critical heat equation in two dimensions with singular data

Nonlinear heat equations in two dimensions with singular initial data are studied. In recent works nonlinearities with exponential growth of Trudinger-Moser type have been shown to manifest critical behavior: well-posedness in the subcritical case and non-existence for certain supercritical data. In this article we propose a specific model nonlinearity with Trudinger-Moser growth for which we obtain surprisingly complete results: a) for initial data strictly below a certain singular threshold function $\widetilde u$ the problem is well-posed, b) for initial data above this threshold function $\widetilde u$, there exists no solution, c) for the singular initial datum $\widetilde u$ there is non-uniqueness. The function $\widetilde u$ is a weak stationary singular solution of the problem, and we show that there exists also a regularizing classical solution with the same initial datum $\widetilde u$.

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arXiv2017

Equivalent and attained version of Hardy's inequality in $\mathbb{R}^n$

We investigate connections between Hardy's inequality in the whole space $\mathbb{R}^n$ and embedding inequalities for Sobolev-Lorentz spaces. In particular, we complete previous results due to [A. Alvino, Sulla diseguaglianza di Sobolev in spazi di Lorentz, (1977)] and [G. Talenti, An inequality between $u^*$ and $|{\rm{grad}} u^*|$, (1992)] by establishing optimal embedding inequalities for the Sobolev-Lorentz quasinorm $\|\nabla\,\cdot\,\|_{p,q}$ also in the range $p < q<\infty$, which remained essentially open since the work of Alvino. Attainability of the best embedding constants is also studied, as well as the limiting case when $q=\infty$. Here, we surprisingly discover that the Hardy inequality is equivalent to the corresponding Sobolev-Marcinkiewicz embedding inequality. Moreover, the latter turns out to be attained by the so-called "ghost" extremal functions of [Brezis-Vázquez, Blow-up solutions of some nonlinear elliptic problems, (1977)], in striking contrast with the Hardy inequality, which is never attained. In this sense, our functional approach seems to be more natural than the classical Sobolev setting, answering a question raised by Brezis and Vázquez.

math.FA↗
arXiv2016

On the Moser-Trudinger inequality in fractional Sobolev-Slobodeckij spaces

We consider the problem of finding the optimal exponent in the Moser-Trudinger inequality \[ \sup \left\{\int_Ω\exp{\left(α\,|u|^{\frac{N}{N-s}}\right)}\,\bigg|\,u \in \widetilde{W}^{s,p}_0(Ω),\,[u]_{W^{s,p}(\mathbb{R}^N)}\leq 1 \right\}< + \infty.\] Here $Ω$ is a bounded domain of $\mathbb{R}^N$ ($N\geq 2$), $s \in (0,1)$, $sp = N$, $\widetilde{W}^{s,p}_0(Ω)$ is a Sobolev-Slobodeckij space, and $[\cdot]_{W^{s,p}(\mathbb{R}^N)}$ is the associated Gagliardo seminorm. We exhibit an explicit exponent $α^*_{s,N}>0$, which does not depend on $Ω$, such that the Moser-Trudinger inequality does not hold true for $α\in (α^*_{s,N},+\infty)$.

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arXiv2006

A sharp Trudinger-Moser type inequality for unbounded domains in $\mathbb{R}^n$

The Trudinger-Moser inequality states that for functions $u \in H_0^{1,n}(Ω)$ ($Ω\subset \mathbb R^n$ a bounded domain) with $\int_Ω|\nabla u|^ndx \le 1$ one has $\int_Ω(e^{α_n|u|^{\frac n{n-1}}}-1)dx \le c |Ω|$, with $c$ independent of $u$. Recently, the second author has shown that for $n = 2$ the bound $c |Ω| $ may be replaced by a uniform constant $d$ independent of $Ω$ if the Dirichlet norm is replaced by the Sobolev norm, i.e. requiring $\int_Ω(|\nabla u|^n + |u|^n)dx \le 1$. We extend here this result to arbitrary dimensions $n > 2$. Also, we prove that for $Ω= \mathbb R^n$ the supremum of $\int_{\mathbb R^n} (e^{α_n|u|^{\frac n{n-1}}}-1)dx$ over all such functions is attained. The proof is based on a blow-up procedure.

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