Vitaly Moroz

Swansea University | SWAN · Department of Mathematics

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 potent...

We study nonnegative optimizers of a Gagliardo-Nirenberg type inequality $$\iint_{\mathbb{R}^N \times \mathbb{R}^N} \frac{|u(x)|^p\,|u(y)|^p}{|x - y|^{N-\alpha}} dx\, dy\le C\Big(\int_{{\mathbb R}^N}|u|^2 dx\Big)^{p\theta} \Big(\int_{{\mathbb R}^N}|u|^q dx\Big)^{2p(1-\theta)/q},$$ that involves the nonlocal Riesz energy with $0<\alpha<N$, $p>\frac{...

We study sub and supersolutions for the $p$-Laplace type elliptic equation of the form $$-\Delta_p u-V|u|^{p-2}u=0\quad\text{in $\Omega$},$$ where $\Omega$ 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 $\O...

We study the Schrödinger–Poisson–Slater equation where p∈(2,3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p\in (2,3)$$\end{document}, λ>0\documentclass[12pt]{minima...

We study normalised solutions of the stationary Gross-Pitaevskii-Poisson (GPP) equation with a defocusing local nonlinear term, $$-\Delta u+\lambda u+|u|^2u =(I_\alpha*|u|^2)u\quad\text{in $\mathbb R^3$},\qquad\int_{\mathbb R^3}u^2dx=\rho^2,$$ where $\rho>0$ is the prescribed mass of the solutions, $\lambda=\lambda_\rho\in\mathbb R$ is an a-priori...

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 potent...

We study asymptotic behaviour of positive ground state solutions of the nonlinear Schrödinger equation $$\begin{aligned} -\Delta u+u=u^{2^*-1}+\lambda u^{q-1} \quad \textrm{in}\, {\mathbb {R}}^N,\qquad \qquad \qquad \qquad \qquad {(P_\lambda )} \end{aligned}$$ - Δ u + u = u 2 ∗ - 1 + λ u q - 1 in R N , ( P λ ) where $$N\ge 3$$ N ≥ 3 is an integer,...

We study asymptotic behaviour of positive ground state solutions of the nonlinear Choquard equation $$ -\Delta u+\varepsilon u=(I_\alpha \ast |u|^{p})|u|^{p-2}u+ |u|^{q-2}u \quad {\rm in} \ \mathbb R^N, $$ where $N\ge 3$ is an integer, $p\in [\frac{N+\alpha}{N}, \frac{N+\alpha}{N-2}]$, $q\in (2,\frac{2N}{N-2}]$, $I_\alpha$ is the Riesz potential an...

We study asymptotic behavior of positive ground state solutions of the nonlinear Kirchhoff equation $$ -\Big(a+b\int_{\mathbb R^N}|\nabla u|^2\Big)\Delta u+ \lambda u= u^{q-1}+ u^{p-1} \quad {\rm in} \ \mathbb R^N, $$ as $\lambda\to 0$ and $\lambda\to +\infty$, where $N=3$ or $N= 4$, $2<q\le p\le 2^*$, $2^*=\frac{2N}{N-2}$ is the Sobolev critical e...

In this paper, we study a class of critical Choquard equations with axisymmetric potentials, $$\begin{aligned} -\Delta u+ V(|x'|,x'')u =\Big (|x|^{-4}*|u|^{2}\Big )u\text{ in } \mathbb {R}^6, \end{aligned}$$where \((x',x'')\in \mathbb {R}^2\times \mathbb {R}^{4}\), \(V(|x'|, x'')\) is a bounded nonnegative function in \(\mathbb {R}^{+}\times \mathb...

In this paper, we study a class of the critical Choquard equations with axisymmetric potentials, $$ -\Delta u+ V(|x'|,x'')u =\Big(|x|^{-4}\ast |u|^{2}\Big)u\hspace{4.14mm}\mbox{in}\hspace{1.14mm} \mathbb{R}^6, $$ 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}...

We study Choquard type equation of the form where $$N\ge 3$$ N ≥ 3 , $$I_\alpha $$ I α is the Riesz potential with $$\alpha \in (0,N)$$ α ∈ ( 0 , N ) , $$p>1$$ p > 1 , $$q>2$$ q > 2 and $$\varepsilon \ge 0$$ ε ≥ 0 . Equations of this type describe collective behaviour of self-interacting many-body systems. The nonlocal nonlinear term represents lon...

We study the Schrödinger–Poisson–Slater equation −Δu+u+λ(I2∗|u|2)u=|u|p−2uin R3,where p∈(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→∞.

We study the Schr\"{o}dinger-Poisson-Slater equation $$-\Delta u + u+\lambda(I_{2}*|u|^2)u=|u|^{p-2}u\quad\text{in $\mathbb R^3$},$$ where $p\in (3,6)$ and $\lambda>0$. 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 $...

We investigate the nonnegative solutions to the nonlinear integral inequality u ≥ I α ∗(( I β ∗ u p ) u q ) a.e. in ${\mathbb R}^{N}$ ℝ N , where α , β ∈ (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...

We study the existence and non-existence of classical solutions for inequalities of type±Δmu≥(Ψ(|x|)⁎up)uq in RN(N≥1). Here, Δm (m≥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...

We study asymptotic behaviour of positive ground state solutions of the nonlinear Schr\"odinger equation $$ -\Delta u+ u=u^{2^*-1}+\lambda u^{q-1} \quad {\rm in} \ \ \mathbb{R}^N, $$ where $N\ge 3$ is an integer, $2^*=\frac{2N}{N-2}$ is the Sobolev critical exponent, $2<q<2^*$ and $\lambda>0$ is a parameter. It is known that as $\lambda\to 0$, afte...

We study Choquard type equation of the form $$-\Delta u +\varepsilon u-(I_{\alpha}*|u|^p)|u|^{p-2}u+|u|^{q-2}u=0\quad in \quad {\mathbb R}^N,\qquad\qquad(P_\varepsilon)$$ where $N\geq3$, $I_\alpha$ is the Riesz potential with $\alpha\in(0,N)$, $p>1$, $q>2$ and $\varepsilon\ge 0$. Equations of this type describe collective behaviour of self-interact...

We investigate the nonnegative solutions to the nonlinear integral inequality $u \ge I_{\alpha}\ast\big((I_\beta \ast u^p)u^q\big)$ a.e. in $\mathbb{R}^N$, where $\alpha, \beta\in (0,N)$, $p, q>0$ and $I_\alpha$, $I_\beta$ denote the Riesz potentials of order $\alpha$ and $\beta$ respectively. Our approach relies on a nonlocal positivity principle...

We study the coupled Hartree system{−Δu+V1(x)u=α1(|x|−4⁎u2)u+β(|x|−4⁎v2)uinRN,−Δv+V2(x)v=α2(|x|−4⁎v2)v+β(|x|−4⁎u2)vinRN, where N≥5, β>max⁡{α1,α2}≥min⁡{α1,α2}>0, and V1,V2∈LN/2(RN)∩Lloc∞(RN) are nonnegative potentials. This system is critical in the sense of the Hardy-Littlewood-Sobolev inequality. For the system with V1=V2=0 we employ moving sphere...

We study the existence and non-existence of classical solutions for inequalities of type $$ \pm \Delta^m u \geq \big(\Psi(|x|)*u^p\big)u^q \quad\mbox{ in } {\mathbb R}^N (N\geq 1). $$ Here, $\Delta^m$ $(m\geq 1)$ is the polyharmonic operator, $p, q>0$ and $*$ denotes the convolution operator, where $\Psi>0$ is a continuous non-increasing function....

This is an outline of the lectures given at Online Mini-courses in Mathematical Analysis 2020, organised by the University of Padova 14-17 September 2020. Please, email me if you spot any typos or mistakes.

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...

We study the coupled Hartree system $$ \left\{\begin{array}{ll} -\Delta u+ V_1(x)u =\alpha_1\big(|x|^{-4}\ast u^{2}\big)u+\beta \big(|x|^{-4}\ast v^{2}\big)u &\mbox{in}\ \mathbb{R}^N,\\[1mm] -\Delta v+ V_2(x)v =\alpha_2\big(|x|^{-4}\ast v^{2}\big)v +\beta\big(|x|^{-4}\ast u^{2}\big)v &\mbox{in}\ \mathbb{R}^N, \end{array}\right. $$ where $N\geq 5$,...

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...

We consider the model of auto-ignition (thermal explosion) of a free round reactive turbulent jet introduced in [11]. 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-Kamene...

Strain and deformation alter the electronic properties of graphene, offering the possibility to control its transport behavior. The tip of a scanning tunneling microscope is an ideal tool to mechanically perturb the system locally while simultaneously measuring the electronic response. Here we stretch few- and multi-layer graphene membranes support...

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: $\alpha$ characterizing the flow rate and $\lambda$ (Frank-Kamentskii...

We study the asymptotic behaviour of positive groundstate solutions to the quasilinear elliptic equation [Figure not available: see fulltext.]where 1 < p< N, p< q< l< + ∞ and ε> 0 is a small parameter. For ε→ 0 , we give a characterization of asymptotic regimes as a function of the parameters q, l and N. In particular, we show that the behaviour of...

We study the asymptotic behavior of positive groundstate solutions to the quasilinear elliptic equation \begin{equation} -\Delta_{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<p<N $, $p<q<l<+\infty$ and $\varepsilon> 0 $ is a small parameter. For $\varepsilon\rightarrow 0$,...

We survey old and recent results dealing with the existence and properties of solutions to the Choquard type equations -Δu+V(x)u=|x|-(N-α)∗|u|p|u|p-2uinRN,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsi...

We prove scaling invariant Gagliardo-Nirenberg type inequalities of the form $$\|\varphi\|_{L^p(\mathbb{R}^d)}\le C\|\varphi\|_{\dot H^{s}(\mathbb{R}^d)}^{\beta} \left(\iint_{\mathbb{R}^d \times \mathbb{R}^d} \frac{|\varphi (x)|^q\,|\varphi (y)|^q}{|x - y|^{d-\alpha}} dx dy\right)^{\gamma},$$ involving fractional Sobolev norms with $s>0$ and Coulom...

We study the nonlocal Schr\"odinger-Poisson-Slater type equation $$ - \Delta u + (I_\alpha \ast \vert u\vert^p)\vert u\vert^{p - 2} u= \vert u\vert^{q-2}u\quad\text{in \(\mathbb{R}^N\),} $$ where $N\in\mathbb{N}$, $p>1$, $q>1$ and $I_\alpha$ is the Riesz potential of order $\alpha\in(0,N).$ We introduce and study the Coulomb-Sobolev function space...

We study a nonlinear equation in the half-space $\{x_1>0\}$ with a Hardy potential, specifically \[-\Delta u -\frac{\mu}{x_1^2}u+u^p=0\quad\text{in}\quad \mathbb R^n_+,\] where $p>1$ and $-\infty<\mu<1/4$. The admissible boundary behavior of the positive solutions is either $O(x_1^{-2/(p-1)})$ as $x_1\to 0$, or is determined by the solutions of the...

We study semilinear elliptic equations with Hardy potential $\mathrm{(E)} \; -L_\mu u+u^q=0$ in a bounded smooth domain $\Omega\subset \mathbb R^N$. Here $q>1$, $L_\mu=\Delta+\frac{\mu}{\delta_\Omega^2}$ and $\delta_\Omega(x)=\mathrm{dist}(x,\partial\Omega)$. Assuming that $0\leq \mu<C_H(\Omega)$, boundary value problems with measure data and discr...

We prove the existence of a minimal action nodal solution for the quadratic Choquard equation $$ -\Delta u + u = \big(I_\alpha \ast |u|^2\big)u \quad\text{in }\; \mathbb R^N,$$ where $I_\alpha$ is the Riesz potential of order $\alpha\in(0,N)$. The solution is constructed as the limit of minimal action nodal solutions for the nonlinear Choquard equa...

We propose a density functional theory of Thomas-Fermi-Dirac-von Weizs\"acker type to describe the response of a single layer of graphene resting on a dielectric substrate to a point charge or a collection of point charges some distance away from the layer. We formulate a variational setting in which the proposed energy functional admits minimizers...

We consider nonlinear Choquard equation $$ - \Delta u + V u = \bigl(I_\alpha \ast |u|^{\frac{\alpha}{N}+1}\bigr) |u|^{\frac{\alpha}{N}-1} u\quad\text{in (\mathbb{R}^N)},$$ where $N \ge 3$, $V \in L^\infty (\mathbb{R}^N)$ is an external potential and $I_\alpha (x)$ is the Riesz potential of order $\alpha \in (0, N)$. The power $\frac{\alpha}{N}+1$ i...

We consider a generalization of the Gelfand problem arising in a Frank-Kamenetskii theory of thermal explosion. This generalization is a natural extension of the Gelfand problem to two phase materials, where, in contrast to classical Gelfand problem which utilizes single temperature approach, the state of the system is described by two different te...

We study the nonlocal equation $$-\varepsilon^2 \Delta u_\varepsilon + V u_\varepsilon = \varepsilon^{-\alpha} \bigl(I_\alpha \ast \abs{u_\varepsilon}^p\bigr) \abs{u_\varepsilon}^{p - 2} u_\varepsilon\quad\text{in \(\mathbf{R}^N\)}, $$ where $N \ge 1$, $\alpha \in (0, N)$, $I_\alpha (x) = A_\alpha/\abs{x}^{N - \alpha}$ is the Riesz potential and $\...

We prove the existence of a nontrivial solution (u \in H^1 (\R^N)) to the nonlinear Choquard equation [- \Delta u + u = \bigl(I_\alpha \ast F (u)\bigr) F' (u) \quad \text{in (\R^N),}] where (I_\alpha) is a Riesz potential, under almost necessary conditions on the nonlinearity (F) in the spirit of Berestycki and Lions. This solution is a groundstate...

Using a groundstate transformation, we give a new proof of the optimal Stein-Weiss inequality of Herbst [\int_{\R^N} \int_{\R^N} \frac{\varphi (x)}{\abs{x}^\frac{\alpha}{2}} I_\alpha (x - y) \frac{\varphi (y)}{\abs{y}^\frac{\alpha}{2}}\dif x \dif y \le \mathcal{C}_{N,\alpha, 0}\int_{\R^N} \abs{\varphi}^2,] and of its combinations with the Hardy ine...

We consider a semilinear elliptic problem [- \Delta u + u = (I_\alpha \ast \abs{u}^p) \abs{u}^{p - 2} u \quad\text{in (\mathbb{R}^N),}] where (I_\alpha) is a Riesz potential and (p>1). This family of equations includes the Choquard or nonlinear Schr\"odinger-Newton equation. For an optimal range of parameters we prove the existence of a positive gr...

We consider a semilinear elliptic problem with a nonlinear term which is the product of a power and the Riesz potential of a power. This family of equations includes the Choquard or nonlinear Schroedinger--Newton equation. We show that for some values of the parameters the equation does not have nontrivial nonnegative supersolutions in exterior dom...

We study the leading order behaviour of positive solutions of the equation -\Delta u +\varepsilon u-|u|^{p-2}u+|u|^{q-2}u=0,\qquad x\in\R^N, where $N\ge 3$, $q>p>2$ and when $\varepsilon>0$ is a small parameter. We give a complete characterization of all possible asymptotic regimes as a function of $p$, $q$ and $N$. The behavior of solutions depend...

We prove the existence of a family of slow-decay positive solutions of a supercritical elliptic equation with Hardy potentialS0013091513000588_Ueqn1and study the stability and oscillation properties of these solutions. We also show that if the equation on ℝ N has a stable slow-decay positive solution, then for any smooth compact K ⊂ ℝ N a family of...

The existence of positive solutions to -epsilon(2) Delta u + Vu = u(p) in R(N), is proved for small epsilon, where N >= 3, V is a nonnegative continuous potential which has a positive local minimum, and N/N-2 < p < N+2/N-2 . No restriction is imposed on the rate of decay of V; this includes in particular the case where V is compactly supported. To...

On a bounded smooth domain Ω ⊂ ℝN , we study solutions of a semilinear elliptic equation with an exponential nonlinearity and a Hardy potential depending on the distance to ∂ ⊂. We derive global a priori bounds of the Keller-Osserman type. Using a Phragmen-Lindelöf alternative for generalize sub- and super-harmonic functions, we discuss the existen...

We study the existence and nonexistence of positive solutions to a sublinear (p<1) second-order divergence type elliptic equation in unbounded cone-like domains CΩ. We prove the existence of the critical exponent which depends on the geometry of the cone CΩ and the coefficients a of the equation.

We study the existence of stationnary positive solutions for a class of nonlinear Schroedinger equations with a nonnegative continuous potential V. Amongst other results, we prove that if V has a positive local minimum, and if the exponent of the nonlinearity satisfies N/(N-2)<p<(N+2)/(N-2), then for small epsilon the problem admits positive soluti...

We study the problem of the existence and nonexistence of positive solutions to the superlinear second-order divergence type elliptic equation with measurable coefficients −\mathrm{∇} \cdot a \cdot \mathrm{∇}u = u^{p} (*),\ p > 1 , in an unbounded cone-like domain G \subset \mathbb{R}^{N} (N⩾3) . We prove that the critical exponent p^{*}(a\text{,}G...

Semilinear elliptic equations which give rise to solutions blowing up at the boundary are perturbed by a Hardy potential. The size of this potential effects the existence of a certain type of solutions (large solutions): if the potential is too small, then no large solution exists. The presence of the Hardy potential requires a new definition of la...

We study the existence and nonexistence of positive (super)solutions to the nonlinear p-Laplace equation in exterior domains of RN (N⩾2). Here p∈(1,+∞) and μ⩽CH, where CH is the critical Hardy constant. We provide a sharp characterization of the set of (q,σ)∈R2 such that the equation has no positive (super)solutions. The proofs are based on the exp...

We study the existence and nonexistence of positive (super-)solutions to a singular semilinear elliptic equation $$-\nabla\cdot(|x|^A\nabla u)-B|x|^{A-2}u=C|x|^{A-\sigma}u^p$$ in cone--like domains of $\R^N$ ($N\ge 2$), for the full range of parameters $A,B,\sigma,p\in\R$ and $C>0$. We provide a complete characterization of the set of $(p,\sigma)\i...

We study the existence and nonexistence of positive (super) solutions to a semi-linear elliptic equation -Delta u - Ax/vertical bar x vertical bar(2). del u - B/vertical bar x vertical bar(2) u = c/vertical bar x vertical bar(sigma) u(p) in cone-like domains of R-N. On the plane R-2 we determine the set of (p, sigma) such that the equation has no p...

We study solutions of the nonlinear Hammer- stein integral equation with changing-sign kernels by using a variational principle of Ricceri and critical points theory tech- niques. Combining the effects of a sublinear and superlinear nonlinear terms we establish new existence and multiplicity results for the equation. As an application we consider a...

We study positive supersolutions to an elliptic equation $(*)$: $-\Delta u=c|x|^{-s}u^p$, $p,s\in\bf R$ in cone-like domains in $\bf R^N$ ($N\ge 2$). We prove that in the sublinear case $p<1$ there exists a critical exponent $p_*<1$ such that equation $(*)$ has a positive supersolution if and only if $-\infty<p<p_*$. The value of $p_*$ is determine...

We consider the Dirichlet problem for the equation −Δu=αu+m(x)u|u|q−2+g(x,u), where q∈(1,2) and m changes sign. We prove that the Morse critical groups at zero of the energy functional of the problem are trivial. As a consequence, existence and bifurcation of nontrivial solutions of the problem are established.

We consider the semilinear Cauchy problem for a class of pseudo-differential operators generating sub-Markovian semigroups. Solutions of such problems with negative definite nonlinearity play an important role in constructing branching measure-valued processes. We establish local existence and uniqueness of solutions in the context of the Dirichlet...

We consider the integral functional of calculus of variations involving higher order derivatives. It is shown that any local minimum of the functional solves the associated strongly nonlinear elliptic problem in a certain weak sense in spite of the fact that no growth conditions are imposed on the zero order term.

We consider the semilinear boundary value problem for pseudo differential operators generating symmetric Dirichlet forms. Using a variational approach we establish the existence of solutions under some growth assumptions on the nonlinearity. We also develop a certain truncation technique based on the specific properties of Dirichlet forms. Such tec...

In this paper we study a nonlinear Hammerstein integral equation by means of the direct variational method. Under certain "natural" growth conditions on the non-linearity we show that the existence of a local minimum for the energy functional implies the solvability of the original equation. In these settings the energy functional may be non-smooth...

In this work we consider a semilinear Dirichlet boundary-value problem with nonlinearity superlinear at zero without any assumptions of symmetry. Using methods of Morse's theory for isolated critical points, we prove the existence of multiple solutions with various assumptions of asymptotic behaviour of the nonlinearity.

In this paper, using a recent result by J. Saint Raymond ([6]), we improve the three critical points theorem established in [5].

A new variant of the mountain pass theorem based on a 'strong' deformation lemma is presented. Some applications to the existence of non-trivial solutions of nonlinear Hammerstein integral equations are given too.