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

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

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}