arXiv subjects

Vitaly Moroz

Publications and source records attributed to Vitaly Moroz.

At least 19 recordsLinked to original sources

arXiv2024

Ground states of a nonlocal variational problem and Thomas-Fermi limit for the Choquard equation

We study nonnegative optimizers of a Gagliardo-Nirenberg type inequality $$\iint_{\mathbb{R}^N \times \mathbb{R}^N} \frac{|u(x)|^p\,|u(y)|^p}{|x - y|^{N-α}} dx\, dy\le C\Big(\int_{{\mathbb R}^N}|u|^2 dx\Big)^{pθ} \Big(\int_{{\mathbb R}^N}|u|^q dx\Big)^{2p(1-θ)/q},$$ that involves the nonlocal Riesz energy with $0<α \frac{N+α}{N}$, $q>\frac{2Np}{N+α}$ and $θ=\frac{(N+α)q-2Np}{Np(q-2)}$. For $p=2$, the equivalent problem has been studied in connection with the Keller-Segel diffusion-aggregation models in the past few decades. The general case $p\neq 2$ considered here appears in the study of Thomas-Fermi limit regime for the Choquard equations with local repulsion. We establish optimal ranges of parameters for the validity of the above interpolation inequality, discuss the existence and qualitative properties of the nonnegative maximizers, and in some special cases estimate the optimal constant. For $p=2$ it is known that the maximizers are Hölder continuous and compactly supported on a ball. We show that for $p<2$ the maximizers are smooth functions supported on $\mathbb{R}^N$, while for $p>2$ the maximizers consist of a characteristic function of a ball and a nonconstant nonincreasing Hölder continuous function supported on the same ball. We use these qualitative properties of the maximizers to establish the validity of the Thomas-Fermi approximations for the Choquard equations with local repulsion. The results are verified numerically with extensive examples.

math.AP↗

arXiv2024

Thomas-Fermi theory of out-of-plane charge screening in graphene

This paper provides a variational treatment of the effect of external charges on the free charges in an infinite free-standing graphene sheet within the Thomas-Fermi theory. We establish existence, uniqueness and regularity of the energy minimizers corresponding to the free charge densities that screen the effect of an external electrostatic potential at the neutrality point. For the potential due to one or several off-layer point charges, we also prove positivity and a precise universal asymptotic decay rate for the screening charge density, as well as an exact charge cancellation by the graphene sheet. We also treat a simpler case of the non-zero background charge density and establish similar results in that case.

math.AP↗

arXiv2024

On the Phragmén-Lindelöf and the superposition principles for the $p$-Laplacian

We study sub and supersolutions for the $p$-Laplace type elliptic equation of the form $$-Δ_p u-V|u|^{p-2}u=0\quad\text{in $Ω$},$$ where $Ω$ is a radially symmetric domain in ${\mathbb{R}}^N$ and $V(x)\ge 0$ is a continuous potential such that the solutions of the equation satisfy the comparison principle on bounded subdomains of $Ω$. In this work we establish a superposition principle and then use it to develop a version of a Phragmén-Lindelöf comparison principle in the case $p\ge 2$. Moreover, by applying this principle to the case of Hardy-type potentials we recover and improve a number of known lower and upper estimates for sub and supersolutions.

math.AP↗

arXiv2024

Asymptotic profiles of ground state solutions for Choquard equations with a general local perturbation

In this paper, we study the asymptotic behavior of ground state solutions for the nonlinear Choquard equation with a general local perturbation $$ -Δu+\varepsilon u=(I_α\ast |u|^{p})|u|^{p-2}u+ g(u), \quad {\rm in} \ \mathbb R^N, \eqno(P_\varepsilon) $$ where $N\ge 3$ is an integer, $p=\frac{N+α}{N}$, or $\frac{N+α}{N-2}$, $I_α$ is the Riesz potential and $\varepsilon>0$ is a parameter. Under some mild conditions on $g(u)$, we show that as $\varepsilon\to \infty$, after {\em a suitable rescaling} the ground state solutions of $(P_\varepsilon)$ converge to a particular solution of some limit equations, and establish a sharp asymptotic characterisation of such a rescaling, which depend in a non-trivial way on the asymptotic behavior of the function $g(s)$ at infinity and the space dimension $N$. Based on this study, we also present some results on the existence and asymptotic behaviors of positive normalized solutions of $(P_\varepsilon)$ with the normalization constraint $\int_{\mathbb R^N}|u|^2=a^2$. Particularly, we obtain the asymptotic behavior of positive normalized solutions of such a problem as $a\to 0$ and $a\to \infty$.

math.AP↗

arXiv2023

Normalised solutions and limit profiles of the defocusing Gross-Pitaevskii-Poisson equation

We study normalised solutions of the stationary Gross-Pitaevskii-Poisson (GPP) equation with a defocusing local nonlinear term, $$-Δu+λu+|u|^2u =(I_α*|u|^2)u\quad\text{in $\mathbb R^3$},\qquad\int_{\mathbb R^3}u^2dx=ρ^2,$$ where $ρ^2>0$ is the prescribed mass of the solutions, $λ\in\mathbb R$ is an a-priori unknown Lagrange multiplier, and $I_α(x)=A_α|x|^{3-α}$ is the Riesz potential of order $α\in(0,3)$. When $α=2$ this problem appears in the models of self-gravitating Bose-Einstein condensates, which were proposed in cosmology and astrophysics to describe Cold Dark Matter and Boson Stars. We establish the existence of branches of normalised solutions to the GPP equation, paying special attention to the shape of the associated mass-energy relation curves and to the limit profiles of solutions at the endpoints of these curves. The main novelty in this work is in the derivation of sharp asymptotic estimates on the mass-energy curves. These estimates allow us to show that after appropriate rescalings, the constructed normalized solutions converge either to a ground state of the Choquard equation or to a compactly supported radial ground state of the integral Thomas-Fermi equation. The behaviour of normalized solutions depends sensitively on whether $α$ is greater than, equal to, or less than one.

math.AP↗

arXiv2023

Asymptotic profiles for a nonlinear Schrödinger equation with critical combined powers nonlinearity

We study asymptotic behaviour of positive ground state solutions of the nonlinear Schrödinger equation $$ -Δu+ u=u^{2^*-1}+λu^{q-1} \quad {\rm in} \ \ \mathbb{R}^N, $$ where $N\ge 3$ is an integer, $2^*=\frac{2N}{N-2}$ is the Sobolev critical exponent, $2 0$ is a parameter. It is known that as $λ\to 0$, after a rescaling the ground state solutions of the equation converge to a particular solution of the critical Emden-Fowler equation $-Δu=u^{2^*-1}$. We establish a sharp asymptotic characterisation of such a rescaling, which depends in a non-trivial way on the space dimension $N=3$, $N=4$ or $N\ge 5$.

math.AP↗

arXiv2023

Asymptotic profiles for Choquard equations with combined attractive nonlinearities

We study asymptotic behaviour of positive ground state solutions of the nonlinear Choquard equation $$ -Δu+\varepsilon u=(I_α\ast |u|^{p})|u|^{p-2}u+ |u|^{q-2}u \quad {\rm in} \ \mathbb R^N, $$ where $N\ge 3$ is an integer, $p\in [\frac{N+α}{N}, \frac{N+α}{N-2}]$, $q\in (2,\frac{2N}{N-2}]$, $I_α$ is the Riesz potential and $\varepsilon>0$ is a parameter. We show that as $\varepsilon\to 0$ (resp. $\varepsilon\to \infty$), after a suitable rescaling the ground state solutions of $(P_\varepsilon)$ converge in $H^1(\mathbb R^N)$ to a particular solution of some limit equations. We also establish a sharp asymptotic characterisation of such a rescaling, and the exact asymptotic behaviours of $u_\varepsilon(0), \|\nabla u_\varepsilon\|_2^2, \|u_\varepsilon\|_2^2, \int_{\mathbb R^N}(I_α\ast |u_\varepsilon|^p)|u_\varepsilon|^p$ and $\|u_\varepsilon\|_q^q$, which depend in a non-trivial way on the exponents $p, q$ and the space dimension $N$. We also discuss a connection of our results with an associated mass constrained problem with normalization constraint $\int_{\mathbb R^N}|u|^2=c^2$. As a consequence of the main results, we obtain the existence, multiplicity and exact asymptotic behaviour of positive normalized solutions of such a problem as $c\to 0$ and $c\to \infty$.

math.AP↗

arXiv2022

Asymptotic profiles for a nonlinear Kirchhoff equation with combined powers nonlinearity

We study asymptotic behavior of positive ground state solutions of the nonlinear Kirchhoff equation $$ -\Big(a+b\int_{\mathbb R^N}|\nabla u|^2\Big)Δu+ λu= u^{q-1}+ u^{p-1} \quad {\rm in} \ \mathbb R^N, $$ as $λ\to 0$ and $λ\to +\infty$, where $N=3$ or $N= 4$, $2 0$, $b\ge 0$ are constants and $λ>0$ is a parameter. In particular, we prove that in the case $2<q<p=2^*$, as $λ\to 0$, after a suitable rescaling the ground state solutions of the problem converge to the unique positive solution of the equation $-Δu+u=u^{q-1}$ and as $λ\to +\infty$, after another rescaling the ground state solutions of the problem converge to a particular solution of the critical Emden-Fowler equation $-Δu=u^{2^*-1}$. We establish a sharp asymptotic characterisation of such rescalings, which depends in a non-trivial way on the space dimension $N=3$ and $N= 4$. We also discuss a connection of our results with a mass constrained problem associated to the Kirchhoff equation with the mass normalization constraint $\int_{\mathbb R^N}|u|^2=c^2$.

math.AP↗

arXiv2022

Construction of infinitely many solutions for a critical Choquard equation via local Pohožaev identities

In this paper, we study a class of the critical Choquard equations with axisymmetric potentials, $$ -Δu+ V(|x'|,x'')u =\Big(|x|^{-4}\ast |u|^{2}\Big)u\hspace{4.14mm}\mbox{in}\hspace{1.14mm} \mathbb{R}^6, $$ where $(x',x'')\in \mathbb{R}^2\times\mathbb{R}^{4}$, $V(|x'|, x'')$ is a bounded nonnegative function in $\mathbb{R}^{+}\times\mathbb{R}^{4}$, and $*$ stands for the standard convolution. The equation is critical in the sense of the Hardy-Littlewood-Sobolev inequality. By applying a finite dimensional reduction argument and developing novel local Pohožaev identities, we prove that if the function $r^2V(r,x'')$ has a topologically nontrivial critical point then the problem admits infinitely many solutions with arbitrary large energies.

math.AP↗

arXiv2022

Limit profiles for singularly perturbed Choquard equations with local repulsion

We study Choquard type equation of the form $$-Δu +\varepsilon u-(I_α*|u|^p)|u|^{p-2}u+|u|^{q-2}u=0\quad in \quad {\mathbb R}^N,\qquad\qquad(P_\varepsilon)$$ where $N\geq3$, $I_α$ is the Riesz potential with $α\in(0,N)$, $p>1$, $q>2$ and $\varepsilon\ge 0$. Equations of this type describe collective behaviour of self-interacting many-body systems. The nonlocal nonlinear term represents long-range attraction while the local nonlinear term represents short-range repulsion. In the first part of the paper for a nearly optimal range of parameters we prove the existence and study regularity and qualitative properties of positive groundstates of $(P_0)$ and of $(P_\varepsilon)$ with $\varepsilon>0$. We also study the existence of a compactly supported groundstate for an integral Thomas-Fermi type equation associated to $(P_\varepsilon)$. In the second part of the paper, for $\varepsilon\to 0$ we identify six different asymptotic regimes and provide a characterisation of the limit profiles of the groundstates of $(P_\varepsilon)$ in each of the regimes. We also outline three different asymptotic regimes in the case $\varepsilon\to\infty$. In one of the asymptotic regimes positive groundstates of $(P_\varepsilon)$ converge to a compactly supported Thomas-Fermi limit profile. This is a new and purely nonlocal phenomenon that can not be observed in the local prototype case of $(P_\varepsilon)$ with $α=0$. In particular, this provides a justification for the Thomas-Fermi approximation in astrophysical models of self-gravitating Bose-Einstein condensate.

math.AP↗

arXiv2022

Asymptotic profile of ground states for the Schrödinger-Poisson-Slater equation

We study the Schrödinger-Poisson-Slater equation $$-Δu + u+λ(I_{2}*|u|^2)u=|u|^{p-2}u\quad\text{in $\mathbb R^3$},$$ where $p\in (3,6)$ and $λ>0$. By using direct variational analysis based on the comparison of the ground state energy levels, we obtain a characterization of the limit profile of the positive ground states for $λ\to \infty$.

math.AP↗

arXiv2021

Nonlinear Inequalities with Double Riesz Potentials

We investigate the nonnegative solutions to the nonlinear integral inequality $u \ge I_α\ast\big((I_β\ast u^p)u^q\big)$ a.e. in $\mathbb{R}^N$, where $α, β\in (0,N)$, $p, q>0$ and $I_α$, $I_β$ denote the Riesz potentials of order $α$ and $β$ respectively. Our approach relies on a nonlocal positivity principle which allows us to derive optimal ranges for the parameters $α$, $β$, $p$ and $q$ to describe the existence and the nonexistence of a solution. The optimal decay at infinity for such solutions is also discussed.

math.AP↗

arXiv2021

Polyharmonic inequalities with nonlocal terms

We study the existence and non-existence of classical solutions for inequalities of type $$ \pm Δ^m u \geq \big(Ψ(|x|)*u^p\big)u^q \quad\mbox{ in } {\mathbb R}^N (N\geq 1). $$ Here, $Δ^m$ $(m\geq 1)$ is the polyharmonic operator, $p, q>0$ and $*$ denotes the convolution operator, where $Ψ>0$ is a continuous non-increasing function. We devise new methods to deduce that solutions of the above inequalities satisfy the poly-superharmonic property. This further allows us to obtain various Liouville type results. Our study is also extended to the case of systems of simultaneous inequalities.

math.AP↗

arXiv2020

Gelfand problem on a large spherical cap

We study the behaviour of the minimal solution to the Gelfand problem on a spherical cap under the Dirichlet boundary conditions. The asymptotic behaviour of the solution is discussed as the cap approaches the whole sphere. The results are based on the sharp estimate of the torsion function of the spherical cap in terms of the principle eigenvalue which we derive in this work.

math.AP↗

arXiv2020

High energy positive solutions for a coupled Hartree system with Hardy-Littlewood-Sobolev critical exponents

We study the coupled Hartree system $$ \left\{\begin{array}{ll} -Δu+ V_1(x)u =α_1\big(|x|^{-4}\ast u^{2}\big)u+β\big(|x|^{-4}\ast v^{2}\big)u &\mbox{in}\ \mathbb{R}^N,\\[1mm] -Δv+ V_2(x)v =α_2\big(|x|^{-4}\ast v^{2}\big)v +β\big(|x|^{-4}\ast u^{2}\big)v &\mbox{in}\ \mathbb{R}^N, \end{array}\right. $$ where $N\geq 5$, $β>\max\{α_1,α_2\}\geq\min\{α_1,α_2\}>0$, and $V_1,\,V_2\in L^{N/2}(\mathbb{R}^N)\cap L_{\text{loc}}^{\infty}(\mathbb{R}^N)$ are nonnegative potentials. This system is critical in the sense of the Hardy-Littlewood-Sobolev inequality. For the system with $V_1=V_2=0$ we employ moving sphere arguments in integral form to classify positive solutions and to prove the uniqueness of positive solutions up to translation and dilation, which is of independent interest. Then using the uniqueness property, we establish a nonlocal version of the global compactness lemma and prove the existence of a high energy positive solution for the system assuming that $|V_1|_{L^{N/2}(\mathbb{R}^N)}+|V_2|_{L^{N/2}(\mathbb{R}^N)}>0$ is suitably small.

math.AP↗

arXiv2019

Gelfand-type problem for turbulent jets

We consider the model of auto-ignition (thermal explosion) of a free round reactive turbulent jet. This model falls into the general class of Gelfand-type problems and constitutes a boundary value problem for a certain semi-linear elliptic equation that depends on two parameters: $α$ characterizing the flow rate and $λ$ (Frank-Kamentskii parameter) characterizing the strength of the reaction. Similarly to the classical Gelfand problem, this equation admits a solution when the Frank-Kametskii parameter $λ$ does not exceed some critical value $λ^*(α)$ and admits no solutions for larger values of $λ$. We obtain the sharp asymptotic behavior of the critical Frank-Kamenetskii parameter in the strong flow limit ($α\gg1$). We also provide a detailed description of the extremal solution (i.e., the solution corresponding to $λ^*$) in this regime.

math.AP↗

arXiv2019

Groundstate asymptotics for a class of singularly perturbed $p$-Laplacian problems in $\mathbb {R}^N$

We study the asymptotic behavior of positive groundstate solutions to the quasilinear elliptic equation \begin{equation} -Δ_{p} u + \varepsilon u^{p-1} - u^{q-1} +u^{\mathit{l}-1} = 0 \qquad \text{in} \quad \mathbb{R}^{N}, \end{equation} where $1 0 $ is a small parameter. For $\varepsilon\rightarrow 0$, we give a characterisation of asymptotic regimes as a function of the parameters $q$, $l$ and $N$. In particular, we show that the behavior of the groundstates is sensitive to whether $q$ is less than, equal to, or greater than the critical Sobolev exponent $p^{*} :=\frac{pN}{N-p}$.

math.AP↗

arXiv2017

Sharp Gagliardo-Nirenberg inequalities in fractional Coulomb-Sobolev spaces

We prove scaling invariant Gagliardo-Nirenberg type inequalities of the form $$\|φ\|_{L^p(\mathbb{R}^d)}\le C\|φ\|_{\dot H^{s}(\mathbb{R}^d)}^β \left(\iint_{\mathbb{R}^d \times \mathbb{R}^d} \frac{|φ(x)|^q\,|φ(y)|^q}{|x - y|^{d-α}} dx dy\right)^γ,$$ involving fractional Sobolev norms with $s>0$ and Coulomb type energies with $0<α 1$.

math.FA↗