Related papers: Conformally invariant equations with negative crit…
We show that nontrivial solutions to higher and fractional order equations with certain nonlinearity are radially symmetric and nonincreasing on geodesic balls in the hyperbolic space $\mathbb{H}^n$ as well as on the entire space…
In this paper we consider lower order perturbations of the critical Lane-Emden system posed on a bounded smooth domain $\Omega \subset \mathbb{R}^N$, with $N \geq3$, inspired by the classical results of Brezis and Nirenberg…
This paper is concerned with damped hyperbolic gradient systems of the form \[ \alpha u_{tt} + u_t = -\nabla V(u) + u_{xx}\,, \] where the spatial domain is the whole real line, the state variable $u$ is multidimensional, $\alpha$ is a…
We consider the nonlinear Schr\"odinger equation $iu_t + \Delta u= \lambda |u|^{\frac {2} {N}} u $ in all dimensions $N\ge 1$, where $\lambda \in {\mathbb C}$ and $\Im \lambda \le 0$. We construct a class of initial values for which the…
We prove that every conformal vector field on the complex hyperbolic space $\mathbb{C}H^n$ is Killing for all $n\ge 2$. Although this rigidity is classically known, our proof is entirely different in nature: it is local, analytic, and fully…
In this paper, we study the large-time behavior of solutions to a class of partially dissipative linear hyperbolic systems with applications in velocity-jump processes in several dimensions. Given integers $n,d\ge 1$, let $\mathbf…
We study both existence and nonexistence of nonnegative solutions for nonlinear elliptic problems with singular lower order terms that have natural growth with respect to the gradient, whose model is $$ \begin{cases} -\Delta u +…
We study a semilinear equation involving the fractional Laplacian on the hyperbolic space $\mathbb{H}^n$. Unlike in conformally compact Einstein manifolds, the fractional Laplacian on $\mathbb{H}^n$ does not enjoy conformal covariance. By…
We prove that the following pointwise inequality holds \begin{equation*} -\Delta u \ge \sqrt\frac{2}{(p+1)-c_n} |x|^{\frac{a}{2}} u^{\frac{p+1}{2}} + \frac{2}{n-4} \frac{|\nabla u|^2}{u} \ \ \text{in}\ \ \mathbb{R}^n \end{equation*} where…
Under suitable growth and coercivity conditions on the nonlinear damping operator $g$ which ensure non-resonance, we estimate the ultimate bound of the energy of the general solution to the equation $\ddot{u}(t) + Au(t) +…
We consider the following quasilinear Schr\"{o}dinger equations of the form \begin{equation*} \triangle u-\varepsilon V(x)u+u\triangle u^2+u^{p}=0,\ u>0\ \mbox{in}\ \mathbb{R}^N\ \mbox{and}\ \underset{|x|\rightarrow \infty}{\lim} u(x)=0,…
We study first-order symmetrizable hyperbolic $N\times N$ systems in a spacetime cylinder whose lateral boundary is totally characteristic. In local coordinates near the boundary at $x=0$, these systems take the form \[ \partial_t u +…
We study the existence of solutions for the nonlinear scalar field equation $$-\Delta u - \frac{(N-2)^2}{4|x|^2} u = g(u), \quad \mbox{in } \mathbb{R}^N \setminus \{0\},$$ where the potential $-\frac{(N-2)^2}{4|x|^2}$ is the critical Hardy…
We are interested in the qualitative properties of positive entire solutions $u \in C^4 (\mathbb{R}^n \backslash \{0\})$ of the equation \begin{equation} \label{0.0} \Delta^2 u=u^{\frac{n+4}{n-4}} \;\;\mbox{in $\mathbb{R}^n \backslash…
In this paper we deal with uniqueness of solutions to the following problem \[ \begin{cases} \begin{split} & u_t-\Delta_p u=H(t,x,\nabla u) &\quad \text{in}\quad Q_T,\\ & u (t,x) =0 &\quad \text{on}\quad(0,T)\times \partial \Omega,\\ &…
We present a comprehensive theory of critical spaces for the broad class of quasilinear parabolic evolution equations. The approach is based on maximal $L_p$-regularity in time-weighted function spaces. It is shown that our notion of…
In this paper, without any assumption on $v$ and under extremely mild assumption $u(x)=O(|x|^{K})$ at $\infty$ for some $K\gg1$ arbitrarily large, we prove classification of solutions to the following conformally invariant system with mixed…
We consider the log-perturbed Br\'ezis-Nirenberg problem on the hyperbolic space \begin{align*} \Delta_{\mathbb{B}^N}u+\lambda u +|u|^{p-1}u+\theta u \ln u^2 =0, \ \ \ \ u \in H^1(\mathbb{B}^N), \ u > 0 \ \mbox{in} \ \mathbb{B}^N,…
In this paper, we study the following nonlinear Kirchhoff problem involving critical growth: $$ \left\{% \begin{array}{ll} -(a+b\int_{\Omega}|\nabla u|^2dx)\Delta u=|u|^4u+\lambda|u|^{q-2}u, u=0\ \ \text{on}\ \ \partial\Omega, \end{array}%…
In this paper, we simplify and extend the results of \cite{GZ} to include the case in which $\Om =\R^3$. Let ${[L^2({\mathbb{R}}^3)]^3}$ be the Hilbert space of square integrable functions on ${\mathbb {R}}^3 $ and let ${\mathbb…