Singular solutions for divergence-form elliptic equations involving regular variation theory: Existence and classification

We generalise and sharpen several recent results in the literature regarding the existence and complete classification of the isolated singularities for a broad class of nonlinear elliptic equations of the form \begin{equation} -{\rm div}\,(\mathcal A(|x|) \,|\nabla u|^{p-2} \nabla u)+b(x)\,h(u)=0\quad \text{in } B_1\setminus\{0\}, \end{equation} where $B_r$ denotes the open ball with radius $r>0$ centred at zero in $\mathbb{R}^N$ $(N\geq 2)$. We assume that $\mathcal{A} \in C^1(0,1]$, $b\in C(\bar{B_1}\setminus\{0\})$ and $h\in C[0,\infty)$ are positive functions associated with regularly varying functions of index $\vartheta$, $\sigma$ and $q$ at $0$, $0$ and $\infty$ respectively, satisfying $q>p-1>0$ and $\vartheta-\sigma<p<N+\vartheta$. We prove that the condition $b(x) \,h(\Phi)\not \in L^1(B_{1/2})$ is sharp for the removability of all singularities at zero for the positive solutions of our problem, where $\Phi$ denotes the "fundamental solution" of $-{\rm div}\,(\mathcal A(|x|)\, |\nabla u|^{p-2} \nabla u)=\delta_0$ (the Dirac mass at zero) in $B_1$, subject to $\Phi|_{\partial B_1}=0$. If $b(x) \,h(\Phi)\in L^1(B_{1/2})$, we show that any non-removable singularity at zero for a positive solution to our equation is either weak (i.e., $\lim_{|x|\to 0} u(x)/\Phi(|x|)\in (0,\infty)$) or strong ($\lim_{|x|\to 0} u(x)/\Phi(|x|)=\infty$). The main difficulty and novelty of this paper, for which we develop new techniques, come from the explicit asymptotic behaviour of the strong singularity solutions in the critical case, which had previously remained open even for $\mathcal{A}=1$. We also study the existence and uniqueness of the positive solution to our problem with a prescribed admissible behaviour at zero and a Dirichlet condition on $\partial B_1$

Examples of sharp asymptotic profiles of singular solutions to an elliptic equation with a sign-changing non-linearity

The first two authors [Proc. Lond. Math. Soc. (3) {\bf 114}(1):1--34, 2017] classified the behaviour near zero for all positive solutions of the perturbed elliptic equation with a critical Hardy--Sobolev growth $$-\Delta u=|x|^{-s} u^{2^\star(s)-1} -\mu u^q \hbox{ in }B\setminus\{0\},$$ where $B$ denotes the open unit ball centred at $0$ in $\mathbb{R}^n$ for $n\geq 3$, $s\in (0,2)$, $2^\star(s):=2(n-s)/(n-2)$, $\mu>0$ and $q>1$. For $q\in (1,2^\star-1)$ with $2^\star=2n/(n-2)$, it was shown in the op. cit. that the positive solutions with a non-removable singularity at $0$ could exhibit up to three different singular profiles, although their existence was left open. In the present paper, we settle this question for all three singular profiles in the maximal possible range. As an important novelty for $\mu>0$, we prove that for every $q\in (2^\star(s) -1,2^\star-1)$ there exist infinitely many positive solutions satisfying $|x|^{s/(q-2^\star(s)+1)}u(x)\to \mu^{-1/(q-2^\star(s)+1)}$ as $|x|\to 0$, using a dynamical system approach. Moreover, we show that there exists a positive singular solution with $\liminf_{|x|\to 0} |x|^{(n-2)/2} u(x)=0$ and $\limsup_{|x|\to 0} |x|^{(n-2)/2} u(x)\in (0,\infty)$ if (and only if) $q\in (2^\star-2,2^\star-1)$.Comment: Mathematische Annalen, to appea

On Logics for Coalgebraic Simulation

AbstractWe investigate logics for coalgebraic simulation from a compositional perspective. Specifically, we show that the expressiveness of an inductively-defined language for coalgebras w.r.t. a given notion of simulation comes as a consequence of an expressivity condition between the language constructor used to define the language for coalgebras, and the relator used to define the notion of simulation. This result can be instantiated to obtain Baltag's logics for coalgebraic simulation, as well as a logic which captures simulation on unlabelled probabilistic transition systems. Moreover, our approach is compositional w.r.t. coalgebraic types. This allows us to derive logics which capture other notions of simulation, including trace inclusion on labelled transition systems, and simulation on discrete Markov processes

Proofs of Urysohn's Lemma and the Tietze Extension Theorem via the Cantor function

Urysohn's Lemma is a crucial property of normal spaces that deals with separation of closed sets by continuous functions. It is also a fundamental ingredient in proving the Tietze Extension Theorem, another property of normal spaces that deals with the existence of extensions of continuous functions. Using the Cantor function, we give alternative proofs for Urysohn's Lemma and the Tietze Extension Theorem.Comment: A slightly modified version has been accepted for publication in the Bulletin of the Australian Mathematical Societ

Sharp asymptotic profiles for singular solutions to an elliptic equation with a sign-changing nonlinearity

Given $B_1(0)$ the unit ball of $\mathbb{R}^n$ ($n\geq 3$), we study smooth positive singular solutions $u\in C^2(B_1(0)\setminus \{0\})$ to $-\Delta u=\frac{u^{2^\star(s)-1}}{|x|^s}-\mu u^q$. Here $0< s<2$, $2^\star(s):=2(n-s)/(n-2)$ is critical for Sobolev embeddings, $q>1$ and $\mu> 0$. When $\mu=0$ and $s=0$, the profile at the singularity $0$ was fully described by Caffarelli-Gidas-Spruck. We prove that when $\mu>0$ and $s>0$, besides this profile, two new profiles might occur. We provide a full description of all the singular profiles. Special attention is accorded to solutions such that $\liminf_{x\to 0}|x|^{\frac{n-2}{2}}u(x)=0$ and $\limsup_{x\to 0}|x|^{\frac{n-2}{2}}u(x)\in (0,+\infty)$. The particular case $q=(n+2)/(n-2)$ requires a separate analysis which we also perform

Singular anisotropic elliptic equations with gradient-dependent lower order terms

We prove the existence of a solution to a singular anisotropic elliptic equation in a bounded open subset $\Omega$ of $\mathbb R^N$ with $N\ge 2$, subject to a homogeneous boundary condition: \begin{equation} \label{eq0} \left\{ \begin{array}{ll} \mathcal A u+ \Phi(u,\nabla u)=\Psi(u,\nabla u)+ \mathfrak{B} u \quad& \mbox{in } \Omega,\\ u=0 & \mbox{on } \partial\Omega. \end{array} \right. \end{equation} Here $\mathcal A u=-\sum_{j=1}^N |\partial_j u|^{p_j-2}\partial_j u$ is the anisotropic $\overrightarrow{p}$-Laplace operator, while $\mathfrak B$ is an operator from $W_0^{1,\overrightarrow{p}}(\Omega)$ into $W^{-1,\overrightarrow{p}'}(\Omega)$ satisfying suitable, but general, structural assumptions. $\Phi$ and $\Psi$ are gradient-dependent nonlinearities whose models are the following: \begin{equation*} \label{phi}\Phi(u,\nabla u):=\left(\sum_{j=1}^N \mathfrak{a}_j |\partial_j u|^{p_j}+1\right)|u|^{m-2}u, \quad \Psi(u,\nabla u):=\frac{1}{u}\sum_{j=1}^N |u|^{\theta_j} |\partial_j u|^{q_j}. \end{equation*} We suppose throughout that, for every $1\leq j\leq N$, \begin{equation*}\label{ass} \mathfrak{a}_j\geq 0, \quad \theta_j>0, \quad 0\leq q_j<p_j, \quad 1<p_j,m\quad \mbox{and}\quad p<N, \end{equation*} and we distinguish two cases: 1) for every $1\leq j\leq N$, we have $\theta_j\geq 1$; 2) there exists $1\leq j\leq N$ such that $\theta_j<1$. In this last situation, we look for non-negative solutions of \eqref{eq0}

1930. Documents Regarding the Passage of Soviet Warships Through the Bosphorus and Dardanelles Straits

The article refers to “the problem of the straits” and the entry into the Black Sea of the Soviet warships “Parischkaya Komuna” and “Profintern”, which led to a “diplomatic conflict” between the Soviet Union and the Straits Commission. The incident came under the attention of Romanian diplomats, who feared that “the Soviets were preparing to attack”

Entire solutions blowing up at infinity for semilinear elliptic systems

AbstractWe consider the system Δu=p(x)g(v), Δv=q(x)f(u) in RN, where f,g are positive and non-decreasing functions on (0,∞) satisfying the Keller–Osserman condition and we establish the existence of positive solutions that blow-up at infinity