Related papers: Limit profiles for singularly perturbed Choquard e…
A quantitative regularity theory is developed for weak solutions to the parabolic system $$ \partial_t u-\mathrm{div}\,{\boldsymbol{\mathsf A}}(x,t,Du)=0 \quad\text{in }E_T\subset \mathbb{R}^N\times\mathbb{R}, $$ which features the…
We prove the existence of infinitely many nonnegative solutions to the following nonlocal elliptic partial differential equation involving singularities \begin{align} (-\Delta)_{p(\cdot)}^{s}…
We consider ground states of the $N$ coupled fermionic nonlinear Schr\"{o}dinger systems with the Coulomb potential $V(x)$ in the $L^2$-subcritical case. By studying the associated constraint variational problem, we prove the existence of…
The paper concerns with positive solutions of problems of the type $-\Delta u+a(x)\, u=u^{p-1}+\varepsilon u^{2^*-1}$ in $\Omega\subseteq\mathbb{R}^N$, $N\ge 3$, $2^*={2N\over N-2}$, $2<p<2^*$. Here $\Omega$ can be an exterior domain, i.e.…
This paper is concerned with the following singularly perturbed non-local semi-linear problem \begin{equation} \label{h} \tag{$\ast$} \begin{cases} \varepsilon^2 \Delta u=\frac{m}{\int_{\Omega}e^{u}{\mathrm{d}x}}u e^u\quad…
In this paper we study positive solutions to the following nonlocal system of equations: \begin{equation*} \left\{\begin{aligned} &(-\Delta)^s u = \frac{\alpha}{2_s^*}|u|^{\alpha-2}u|v|^{\beta}+f(x)\;\;\text{in}\;\mathbb{R}^{N},…
We look for nonconstant, positive, radially nondecreasing solutions of the quasilinear equation $-\Delta_p u+u^{p-1}=f(u)$ with $p>2$, in the unit ball $B$ of $\mathbb R^N$, subject to homogeneous Neumann boundary conditions. The…
Asymptotic reductions of a defocusing nonlocal nonlinear Schr\"{o}dinger model in $(3+1)$-dimensions, in both Cartesian and cylindrical geometry, are presented. First, at an intermediate stage, a Boussinesq equation is derived, and then its…
We study the large time behavior of solutions to the Cauchy problem for the quasilinear absorption-diffusion equation $$ \partial_tu=\Delta u^m-|x|^{\sigma}u^p, \quad (x,t)\in\real^N\times(0,\infty), $$ with exponents $p>m>1$ and $\sigma>0$…
We discuss the existence of ground state solutions for the Choquard equation $$-\Delta u=(I_\alpha*F(u))F'(u)\quad\quad\quad\text{in }\mathbb R^N.$$ We prove the existence of solutions under general hypotheses, investigating in particular…
In this paper, we consider the following nonlinear parabolic equation \[ \partial_{t}u\,=\,\sum_{i=1}^{n}\partial_{x_{i}}\left[(\vert u_{x_{i}}\vert-\delta_{i})_{+}^{p-1}\frac{u_{x_{i}}}{\vert…
In this work, we study the existence of various classes of standing waves for a nonlinear Schr\"odinger system with quadratic interaction, along with a harmonic or partially harmonic potential. We establish the existence of ground-state…
In this paper, we show existence of \textit{continuums} of positive solutions for non-local quasilinear problems with strongly-singular reaction term on a bounded domain in $\mathbb{R}^N$ with $N \geq 2$. We approached non-autonomous and…
Optimal estimates on asymptotic behaviors of weak solutions both at the origin and at the infinity are obtained to the following quasilinear elliptic equations \begin{eqnarray*} -\Delta_{p}u-\frac{\mu}{|x|^{p}}|u|^{p-2}u+m|u|^{p-2}u=f(u), &…
We consider a class of nonlinear Schroedinger equation in three space dimensions with an attractive potential. The nonlinearity is local but rather general encompassing for the first time both subcritical and supercritical (in $L^2$)…
We are concerned with solvability of nonlinear systems involving a discrete singular $\phi$-Laplacian operator of type \begin{equation*} u \mapsto \Delta\left[\phi(\Delta u(n-1))\right] \qquad (n\in \{1, \dots, T\}), \end{equation*}…
We study the monotonicity and one-dimensional symmetry of positive solutions to the problem $-\Delta_p u = f(u)$ in $\mathbb{R}^N_+$ under zero Dirichlet boundary condition, where $p>1$ and $f:(0,+\infty)\to\mathbb{R}$ is a locally…
We study the behavior near the origin of $C^2$ positive solutions $u(x)$ and $v(x)$ of the system $0\leq -\Delta u\leq f(v)$ $0\leq -\Delta v\leq g(u)$ in $B_1(0)\backslash\{0\}$ where $f,g:(0,\infty)\to (0,\infty)$ are continuous…
This article focuses on the study of the existence, multiplicity and concentration behavior of ground states as well as the qualitative aspects of positive solutions for a $(p, N)$-Laplace Schr\"{o}dinger equation with logarithmic…
We prove a quantitative version of the isoperimetric inequality for a non local perimeter of Minkowski type. We also apply this result to study isoperimetric problems with repulsive interaction terms, under convexity constraints. We show…