In this addendum we fill a gap in a proof and we correct some results appearing in [12]. In the original paper [12] we classified positive solutions for the following equation \begin{equation*} \Delta_{p}u+K(r) u^{\sigma-1}=0 \end{equation*} where $r=|x|$, $x \in \mathbb R^n$, $n >p>1$, $\sigma ={n p}/({n-p})$ and $K(r)$ is a function strictly positive and bounded. In fact [12] had two main purposes. First, to establish asymptotic conditions which are sufficient for the existence of ground states with fast decay and to classify regular and singular solutions: these results are correct but need some non-trivial further explanations. Second to establish some computable conditions on $K$ which are sufficient to obtain multiplicity of ground states with fast decay in a non-perturbation context. Also in this case the original argument contained a flaw: here we correct the assumptions of [12] by performing a new nontrivial construction. A third purpose of this addendum is to generalize results of [12] to a slightly more general equation \begin{equation*} \Delta_p u+ r^{\delta}K(r) u^{\sigma(\delta)-1}=0 \end{equation*} where $\delta>-p$, and $\sigma(\delta) ={p(n+\delta)}/({n-p})$.

We consider the following quasilinear elliptic equation with critical Sobolev and Hardy-Sobolev exponents: \[ \begin{cases} \displaystyle -\sum\limits_{i,j=1}^ND_j(b_{ij}(v)D_iv)+\frac{1}{2}\sum\limits_{i,j=1}^N b_{ij}'(v)D_ivD_jv \\ \displaystyle \qquad\quad =\frac{|v|^{2^*_sq-2}v}{|x|^s}+\mu|v|^{2^*q-2}v+a(x)|v|^{2q-2}v &\hbox{in }\Omega,\\ v=0 &\hbox{on } \partial\Omega, \end{cases} \] where $b_{ij}\in C^1(\mathbb{R},\mathbb{R})$ satisfies the growth condition $|b_{ij}(t)|\sim|t|^{2(q-1)}$ at infinity, $q\geq1$, $\mu\geq0$, $0< s< 2$, $2^*_s={2(N-s)}/({N-2})$, $2^*={2N}/({N-2})$, $0\in\overline{\Omega}$ and $\Omega$ is a bounded domain in $\mathbb{R}^N$. In this paper, we will investigate the effects of the lower order terms $a(x)|v|^{2q-2}v$ and the growth of $b_{ij}(v)$ at infinity on the existence of infinitely many solutions for the above equations.

In this paper we introduce another type of expansiveness, called mild expansiveness, for homeomorphisms on compact metric spaces, and present a topological version of the spectral decomposition theorem by Smale [17] and Bowen [5]. Moreover we study the topological stability of mild expansive homeomorphisms, and characterize the mild expansive diffeomorphisms on compact smooth manifolds using the notion of $\Omega$-stability.

In this paper, we introduce the definition of a $C_0$-semigroup on a time scale, which unifies the continuous, discrete and other cases which lie between them. Also, it extends the classical theory of operator semigroups to the quantum case. We study the relationship between the semigroup and its infinitesimal generator. We apply our theory to study the homogeneous and non homogeneous abstract Cauchy problem in Banach and Fréchet spaces.

We prove an existence result for a semilinear elliptic equation with superlinear and asymmetric nonlinearity. The asymmetry that we consider is of the type: linear at $-\infty$ and superlinear at $+\infty$. To obtain these results we apply a Linking Theorem.

In this paper we establish the existence of bounded solutions to a strongly nonlinear elliptic problem of the form $$ -\mathop{\rm div}\mathcal{A}(x,u,{\nabla}u)+g(x,u,\nabla u)= f \quad\text{in }{\Omega}, $$ with $u\in W^1_0L_\varphi({\Omega})\cap L^{\infty}(\Omega)$, where $$ \mathcal{A}(x,s,\xi)\cdot\xi\geq \overline{\varphi}_{x}^{-1} (\varphi(x,h(|s|)))\varphi(x,|\xi|), $$ $h\colon {\mathbb{R}^+} \to \mathopen ]0,1] $ is a continuous decreasing function with unbounded primitive and $g$ is a non-linearity satisfying $|g(x,s,\xi)|\leq\beta(s)\varphi(x,|\xi|)$. We assume the $\Delta_{2}$-condition on the Musielak function $\varphi$.

We investigate under which conditions the space of idempotent measures is an absolute retract and the idempotent barycenter map is soft.

A class of generalized Schrödinger problems in a bounded domain is studied. A complete overview of the set of solutions is provided, depending on the values assumed by parameters involved in the problem. In order to obtain the results, we combine monotony, bifurcation and variational methods.

The deterministic analog of the Markov property of a time-homogeneous Markov process is the semigroup property of solutions of an autonomous differential equation. The semigroup property arises naturally when the solutions of a differential equation are unique, and leads to a semiflow. We prove an abstract result on the measurable selection of a semiflow for the situations without uniqueness. We outline applications to ODEs, PDEs, differential inclusions, etc. Our proof of the semiflow selection theorem is motivated by N.V. Krylov's Markov selection theorem. To accentuate this connection, we include a new version of the Markov selection theorem related to more recent papers of Flandoli, Romito and Goldys et al.

We establish a new limiting result which extends the famous Whyburn's limit theorem. As applications, we study the existence and multiplicity of one-sign or sign-changing solutions with a prescribed number of simple zeros for the following problem \begin{equation*} \begin{cases} -\mathcal{M}_{\lambda,\Lambda}^+\left(D^2 u\right)=\mu f(u) &\text{in } \Omega,\\ u=0&\text{on } \partial\Omega, \end{cases} \end{equation*} where $\mathcal{M}_{\lambda,\Lambda}^+$ denotes the Pucci extremal operator. Combining bifurcation approach with our generalized limit theorem, we determine the range of parameter $\mu$ in which the above problem has one or multiple one-sign or sign-changing solutions according to the behaviors of $f$ at $0$ and $\infty$, and whether $f$ satisfies the signum condition $f(s)s> 0$ for $s\neq0$.

In this paper, we consider the following nonhomogeneous Kirchhoff-Schrödinger equation: $$ m\bigg(\int_{\mathbb{R}^{2}}|\nabla u|^2\,{d}x +\int_{\mathbb{R}^{2}}V(|x|)u^2\,{d}x \bigg) [-\Delta u + V(|x|)u] = Q(|x|)f(u) + \varepsilon h(x), $$ for $ x\in\mathbb{R}^2$, where $m$, $ V$, $ Q$ and $f$ are continuous functions, $\varepsilon$ is a small parameter and $h\neq 0$. When $f$ has exponential growth by means of a Trudinger-Moser type inequality, the Mountain Pass Theorem and Ekeland's Variational Principle in weighted Sobolev spaces are applied in order to establish the existence of at least two weak solutions for this equation.

In this paper, a singular one-dimensional viscoelastic system with a nonlinear source term, nonlocal boundary condition and damping terms is considered. We prove the existence of a global solution using the potential-well theory. Furthermore, by constructing Lyapunov functional combined with the perturbed energy method, the general decay result is proved.

We deal with an aggregate version of the KKM theorem. Given $n$ KKM families of special type on an $( n-1)$-dimensional simplex, we show that it is possible to choose a single element from every KKM family to get a KKM family on that simplex. We also introduce and study function KKM families as surrogates of KKM families on simplexes. We show a function version of the KKM theorem. The Coincidence Theorem is our main result, Brouwer's fixed is a special cases of that theorem.

In this paper, we address an open question regarding common fixed points of a family of non-self mappings. More precisely, we employ the Krasnosel'skiĭ-Mann algorithm to approximate common fixed points of a countable family of Lipschitzian non-self mappings in Banach spaces under different conditions.

This paper presents a generalized epidemic model with discontinuous treatment strategies and time-varying delays. Under the concept of Filippov solution, by applying the differential inclusions and the topological degree theory in set-valued analysis, we employ a novel argument to establish new results on the existence of the periodic solutions for the considered epidemic model. After that, we derive some criteria on the uniqueness, global exponential stability of the considered epidemic model and convergence of the corresponding autonomous case of the considered epidemic model, in terms of nonsmooth analysis theory with the Lyapunov-like approach. Our results extend previous works on the epidemic model to the discontinuous cases, some corresponding results in the literature can be enriched and extended. Finally, typical examples and the corresponding numerical simulations have been carried out to support the analytic findings.