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Vitaly Moroz

Publications and source records attributed to Vitaly Moroz.

At least 37 records · Page 2Linked to original sources
arXiv2016

Groundstates and radial solutions to nonlinear Schrödinger-Poisson-Slater equations at the critical frequency

We study the nonlocal Schrödinger-Poisson-Slater type equation $$ - Δu + (I_α\ast |u|^p)|u|^{p - 2} u= |u|^{q-2}u\quad\text{in \(\mathbb{R}^N\),} $$ where $N\in\mathbb{N}$, $p>1$, $q>1$ and $I_α$ is the Riesz potential of order $α\in(0,N).$ We introduce and study the Coulomb-Sobolev function space which is natural for the energy functional of the problem and we establish a family of associated optimal interpolation inequalities. We prove existence of optimizers for the inequalities, which implies the existence of solutions to the equation for a certain range of the parameters. We also study regularity and some qualitative properties of solutions. Finally, we derive radial Strauss type estimates and use them to prove the existence of radial solutions to the equation in a range of parameters which is in general wider than the range of existence parameters obtained via interpolation inequalities.

math.AP↗
arXiv2016

Boundary singularities of solutions of semilinear elliptic equations in the half-space with a Hardy potential

We study a nonlinear equation in the half-space $\{x_1>0\}$ with a Hardy potential, specifically \[-Δu -\fracμ{x_1^2}u+u^p=0\quad\text{in}\quad \mathbb R^n_+,\] where $p>1$ and $-\infty<μ<1/4$. 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 linear problem $-Δh -\fracμ{x_1^2}h=0$. In the first part we study in full detail the separable solutions of the linear equations for the whole range of $μ$. In the second part, by means of sub and supersolutions we construct separable solutions of the nonlinear problem which behave like $O(x_1^{-2/(p-1)})$ near the origin and which, away from the origin have exactly the same asymptotic behavior as the separable solutions of the linear problem. In the last part we construct solutions that behave like $O(x_1^{-2/(p-1)})$ at some prescribed parts of the boundary, while at the rest of the boundary the solutions decay or blowup at a slower rate determined by the linear part of the equation.

math.AP↗
arXiv2016

Moderate solutions of semilinear elliptic equations with Hardy potential under minimal restrictions on the potential

We study semilinear elliptic equations with Hardy potential $\mathrm{(E)} \; -L_μu+u^q=0$ in a bounded smooth domain $Ω\subset \mathbb R^N$. Here $q>1$, $L_μ=Δ+\fracμ{δ_Ω^2}$ and $δ_Ω(x)=\mathrm{dist}(x,\partialΩ)$. Assuming that $0\leq μ<C_H(Ω)$, boundary value problems with measure data and discrete boundary singularities for positive solutions of $\mathrm{(E)}$ have been studied earlier. In the present paper we study these problems, in arbitrary domains, assuming only $-\infty<μ<1/4$, even if $C_H(Ω)<1/4$. We recall that $C_H(Ω)\leq 1/4$ and, in general, strict inequality holds. The key to our study is the fact that, if $μ<1/4$ then in smooth domains there exist local $L_μ$-superharmonic functions in a neighborhood of $\partialΩ$ (even if $C_H(Ω)<1/4$). Using this fact we extend the notion of normalized boundary trace to arbitrary domains, provided that $μ<1/4$. Further we study the b.v.p. with normalized boundary trace $ν$ in the space of positive finite measures on $\partialΩ$. We show that existence depends on two critical values of the exponent $q$ and discuss the question of uniqueness. Part of the paper is devoted to the study of the linear operator: properties of local $L_μ$ subharmonic and superharmonic functions and the related notion of moderate solutions.

math.AP↗
arXiv2016

A guide to the Choquard equation

We survey old and recent results dealing with the existence and properties of solutions to the Choquard type equations $$ -Δu + V(x)u = \bigl(|x|^{-(N-α)} * |u|^p\bigr)|u|^{p - 2} u \qquad \text{in $\mathbb{R}^N$}, $$ and some of its variants and extensions.

math.AP↗
arXiv2016

Least action nodal solutions for the quadratic Choquard equation

We prove the existence of a minimal action nodal solution for the quadratic Choquard equation $$ -Δu + u = \big(I_α\ast |u|^2\big)u \quad\text{in }\; \mathbb R^N,$$ where $I_α$ is the Riesz potential of order $α\in(0,N)$. The solution is constructed as the limit of minimal action nodal solutions for the nonlinear Choquard equations $$ -Δu + u = \big(I_α\ast |u|^p\big)|u|^{p-2}u \quad\text{in }\; \mathbb R^N$$ when $p\searrow 2$. The existence of minimal action nodal solutions for $p>2$ can be proved using a variational minimax procedure over Nehari nodal set. No minimal action nodal solutions exist when $p<2$.

math.AP↗
arXiv2015

Orbital-free density functional theory of out-of-plane charge screening in graphene

We propose a density functional theory of Thomas-Fermi-Dirac-von Weizsäcker type to describe the response of a single layer of graphene resting on a dielectric substrate to a point charge or a collection of charges some distance away from the layer. We formulate a variational setting in which the proposed energy functional admits minimizers, both in the case of free graphene layers and under back-gating. We further provide conditions under which those minimizers are unique and correspond to configurations consisting of inhomogeneous density profiles of charge carrier of only one type. The associated Euler-Lagrange equation for the charge density is also obtained, and uniqueness, regularity and decay of the minimizers are proved under general conditions. In addition, a bifurcation from zero to non-zero response at a finite threshold value of the external charge is proved.

math-ph↗
arXiv2014

Groundstates of nonlinear Choquard equations: Hardy-Littlewood-Sobolev critical exponent

We consider nonlinear Choquard equation $$ - Δu + V u = \bigl(I_α\ast |u|^{\fracα{N}+1}\bigr) |u|^{\fracα{N}-1} u\quad\text{in (\mathbb{R}^N)},$$ where $N \ge 3$, $V \in L^\infty (\mathbb{R}^N)$ is an external potential and $I_α(x)$ is the Riesz potential of order $α\in (0, N)$. The power $\fracα{N}+1$ in the nonlocal part of the equation is critical with respect to the Hardy-Littlewood-Sobolev inequality. As a consequence, in the associated minimization problem a loss of compactness may occur. We prove that if $\liminf_{|x| \to \infty} \bigl(1 - V (x)\bigr)|x|^2 > \frac{N^2 (N - 2)}{4 (N + 1)}$ then the equation has a nontrivial solution. We also discuss some necessary conditions for the existence of a solution. Our considerations are based on a concentration compactness argument and a nonlocal version of Brezis-Lieb lemma.

math.AP↗
arXiv2013

Gelfand type problem for two phase porous media

We consider a generalization of the Gelfand problem arising in Frank-Kamenetskii theory of thermal explosion. This generalization is a natural extension of the Gelfand problem to two phase materials, where, in contrast to the classical Gelfand problem which utilizes single temperature approach, the state of the system is described by two different temperatures. We show that similar to the classical Gelfand problem the thermal explosion occurs exclusively due to the absence of stationary temperature distribution. We also show that the presence of inter-phase heat exchange delays a thermal explosion. Moreover, we prove that in the limit of infinite heat exchange between phases the problem of thermal explosion in two phase porous media reduces to the classical Gelfand problem with renormalized constants.

math.AP↗
arXiv2013

Groundstates of nonlinear Choquard equations: existence, qualitative properties and decay asymptotics

We consider a semilinear elliptic problem [- Δu + u = (I_α\ast \abs{u}^p) \abs{u}^{p - 2} u \quad\text{in (\mathbb{R}^N),}] where (I_α) is a Riesz potential and (p>1). This family of equations includes the Choquard or nonlinear Schrödinger-Newton equation. For an optimal range of parameters we prove the existence of a positive groundstate solution of the equation. We also establish regularity and positivity of the groundstates and prove that all positive groundstates are radially symmetric and monotone decaying about some point. Finally, we derive the decay asymptotics at infinity of the groundstates.

math.AP↗
arXiv2012

Existence of groundstates for a class of nonlinear Choquard equations

We prove the existence of a nontrivial solution (u \in H^1 (\R^N)) to the nonlinear Choquard equation [- Δu + u = \bigl(I_α\ast F (u)\bigr) F' (u) \quad \text{in (\R^N),}] where (I_α) is a Riesz potential, under almost necessary conditions on the nonlinearity (F) in the spirit of Berestycki and Lions. This solution is a groundstate; if moreover (F) is even and monotone on ((0,\infty)), then (u) is of constant sign and radially symmetric.

math.AP↗
arXiv2012

Nonlocal Hardy type inequalities with optimal constants and remainder terms

Using a groundstate transformation, we give a new proof of the optimal Stein-Weiss inequality of Herbst [\int_{\R^N} \int_{\R^N} \frac{φ(x)}{\abs{x}^\fracα{2}} I_α(x - y) \frac{φ(y)}{\abs{y}^\fracα{2}}\dif x \dif y \le \mathcal{C}_{N,α, 0}\int_{\R^N} \absφ^2,] and of its combinations with the Hardy inequality by Beckner [\int_{\R^N} \int_{\R^N} \frac{φ(x)}{\abs{x}^\frac{α+ s}{2}} I_α(x - y) \frac{φ(y)}{\abs{y}^\frac{α+ s}{2}}\dif x \dif y \le \mathcal{C}_{N, α, 1} \int_{\R^N} \abs{\nabla φ}^2,] and with the fractional Hardy inequality [\int_{\R^N} \int_{\R^N} \frac{φ(x)}{\abs{x}^\frac{α+ s}{2}} I_α(x - y) \frac{φ(y)}{\abs{y}^\frac{α+ s}{2}}\dif x \dif y \le \mathcal{C}_{N, α, s} \mathcal{D}_{N, s} \int_{\R^N} \int_{\R^N} \frac{\bigabs{φ(x) - φ(y)}^2}{\abs{x-y}^{N+s}}\dif x \dif y] where (I_α) is the Riesz potential, (0 < α< N) and (0 < s < \min(N, 2)). We also prove the optimality of the constants. The method is flexible and yields a sharp expression for the remainder terms in these inequalities.

math.AP↗
arXiv2012

Nonexistence and optimal decay of supersolutions to Choquard equations in exterior domains

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 domains. The same techniques yield optimal decay rates when supersolutions exists.

math.AP↗
arXiv2012

Asymptotic properties of ground states of scalar field equations with a vanishing parameter

We study the leading order behaviour of positive solutions of the equation -Δu +\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 depends sensitively on whether $p$ is less, equal or bigger than the critical Sobolev exponent $p^\ast=\frac{2N}{N-2}$. For $p p^\ast$ the solution asymptotically coincides with the solution of the equation with $\varepsilon=0$. In the most delicate case $p=p^\ast$ the asymptotic behaviour of the solutions is given by a particular solution of the critical Emden--Fowler equation, whose choice depends on $\varepsilon$ in a nontrivial way.

math.AP↗
arXiv2009

Large solutions to semilinear elliptic equations with Hardy potential and exponential nonlinearity

On a bounded smooth domain we study solutions of a semilinear elliptic equation with an exponential nonlinearity and a Hardy potential depending on the distance to the boundary of the domain. We derive global a priori bounds of the Keller-Osserman type. Using a Phragmen-Lindelof alternative for generalized sub and super-harmonic functions we discuss existence, nonexistence and uniqueness of so-called large solutions, i.e., solutions which tend to infinity at the boundary. The approach develops the one used by the same authors for a problem with a power nonlinearity instead of the exponential nonlinearity.

math.AP↗
arXiv2009

Semiclassical stationary states for nonlinear Schroedinger equations with fast decaying potentials

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 solutions which concentrate as epsilon goes to 0 around the local minimum point of V. The novelty is that no restriction is imposed on the rate of decay of V. In particular, we cover the case where V is compactly supported.

math.AP↗
arXiv2007

"Boundary blowup" type sub-solutions to semilinear elliptic equations with Hardy potential

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 large solutions, following the pattern of the associated linear problem. Nonexistence and existence results for different types of solutions will be given. Our considerations are based on a Phragmen-Lindelof type theorem which enables us to classify the solutions and sub-solutions according to their behavior near the boundary. Nonexistence follows from this principle together with the Keller-Osserman upper bound. The existence proofs rely on sub- and super-solution techniques and on estimates for the Hardy constant derived in Marcus, Mizel and Pinchover.

math.AP↗
arXiv2006

Positive solutions to nonlinear p-Laplace equations with Hardy potential in exterior domains

We study the existence and nonexistence of positive (super) solutions to the nonlinear $p$-Laplace equation $$-Δ_p u-\fracμ{|x|^p}u^{p-1}=\frac{C}{|x|^σ}u^q$$ in exterior domains of ${\R}^N$ ($N\ge 2$). Here $p\in(1,+\infty)$ and $μ\le C_H$, where $C_H$ is the critical Hardy constant. We provide a sharp characterization of the set of $(q,σ)\in\R^2$ such that the equation has no positive (super) solutions. The proofs are based on the explicit construction of appropriate barriers and involve the analysis of asymptotic behavior of super-harmonic functions associated to the $p$-Laplace operator with Hardy-type potentials, comparison principles and an improved version of Hardy's inequality in exterior domains. In the context of the $p$-Laplacian we establish the existence and asymptotic behavior of the harmonic functions by means of the generalized Prüfer-Transformation.

math.AP↗