# Vladimir BobkovInstitute of Mathematics, Ufa Federal Research Centre, Russian Academy of Sciences

Vladimir Bobkov

CSc.

## About

49

Publications

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231

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Introduction

Additional affiliations

February 2016 - present

August 2013 - August 2014

November 2011 - present

**Institute of Mathematics with Computer Center of Russian Academy of Sciences**

Position

- Researcher

## Publications

Publications (49)

Using symmetrization techniques, we show that, for every N ≥ 2 N \geq 2 , any second eigenfunction of the fractional Laplacian in the N N -dimensional unit ball with homogeneous Dirichlet conditions is nonradial, and hence its nodal set is an equatorial section of the ball.

We investigate the basis properties of sequences of Fučík eigenfunctions of the one-dimensional Neumann Laplacian. We show that any such sequence is complete in L2(0,π) and a Riesz basis in the subspace of functions with zero mean. Moreover, we provide sufficient assumptions on Fučík eigenvalues which guarantee that the corresponding Fučík eigenfun...

We investigate the basis properties of sequences of Fucik eigenfunctions of the one-dimensional Neumann Laplacian. We show that any such sequence is complete in $L^2(0,\pi)$ and a Riesz basis in the subspace of functions with zero mean. Moreover, we provide sufficient assumptions on Fucik eigenvalues which guarantee that the corresponding Fucik eig...

We establish sufficient assumptions on sequences of Fučík eigenvalues of the one-dimensional Laplacian which guarantee that the corresponding Fučík eigenfunctions form a Riesz basis in L2(0,π).

We provide improved sufficient assumptions on sequences of Fucik eigenvalues of the one-dimensional Dirichlet Laplacian which guarantee that the corresponding Fucik eigenfunctions form a Riesz basis in $L^2(0,\pi)$. For that purpose, we introduce a criterion for a sequence in a Hilbert space to be a Riesz basis.

We investigate the properties of the Cheeger sets of rotationally invariant, bounded domains Ω⊂Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega \subset \mathbb...

We study the existence, multiplicity, and certain qualitative properties of solutions to the zero Dirichlet problem for the equation $-\Delta_p u = \lambda |u|^{p-2}u + a(x)|u|^{q-2}u$ in a bounded domain $\Omega \subset \mathbb{R}^N$, where $1<q<p$, $\lambda\in\mathbb{R}$, and $a$ is a continuous sign-changing weight function. Our primary interest...

We study the zero Dirichlet problem for the equation [Formula: see text] in a bounded domain [Formula: see text], with [Formula: see text]. We investigate the relation between two critical curves on the [Formula: see text]-plane corresponding to the threshold of existence of special classes of positive solutions. In particular, in certain neighborh...

Using symmetrization techniques, we show that, for every $N \geq 2$, any second eigenfunction of the fractional Laplacian in the $N$-dimensional unit ball with homogeneous Dirichlet conditions is nonradial, and hence its nodal set is an equatorial section of the ball.

Let $\mu_2(\Omega)$ be the first positive eigenvalue of the Neumann Laplacian in a bounded domain $\Omega \subset \mathbb{R}^N$. It was proved by Szego for $N=2$ and by Weinberger for $N \geq 2$ that among all equimeasurable domains $\mu_2(\Omega)$ attains its global maximum if $\Omega$ is a ball. In the present work, we develop the approach of Wei...

We establish sufficient assumptions on sequences of Fucik eigenvalues of the one-dimensional Laplacian which guarantee that the corresponding Fucik eigenfunctions form a Riesz basis in $L^2(0,\pi)$.

We study the existence and multiplicity of nonnegative solutions, as well as the behavior of corresponding parameter-dependent branches, to the equation −Δu=(1−u)um−λun in a bounded domain Ω⊂RN endowed with the zero Dirichlet boundary data, where 0<m≤1 and n>0. When λ>0, the obtained solutions can be seen as steady states of the corresponding react...

We obtain a generalization of the Picone inequality which, in combination with the classical Picone inequality, appears to be useful for problems with the ( p , q ) -Laplace-type operators. With its help, as well as with the help of several other known generalized Picone inequalities, we provide some nontrivial facts on the existence and nonexisten...

We study the zero Dirichlet problem for the equation $-\Delta_p u -\Delta_q u = \alpha |u|^{p-2}u+\beta |u|^{q-2}u$ in a bounded domain $\Omega \subset \mathbb{R}^N$, with $1<q<p$. We investigate the relation between two critical curves on the $(\alpha,\beta)$-plane corresponding to the threshold of existence of special classes of positive solution...

We obtain a generalization of the Picone inequality which, in combination with the classical Picone inequality, appears to be useful for problems with the $(p,q)$-Laplace type operators. With its help, as well as with the help of several other known generalized Picone inequalities, we provide some nontrivial facts on the existence and nonexistence...

We study the existence and multiplicity of nonnegative solutions, as well as the behaviour of corresponding parameter-dependent branches, to the equation $-\Delta u = (1-u) u^m - \lambda u^n$ in a bounded domain $\Omega \subset \mathbb{R}^N$ endowed with the zero Dirichlet boundary data, where $0 < m \leq 1$ and $n > 0$. When $\lambda > 0$, the obt...

Let [Formula: see text] be an arithmetic function with [Formula: see text] and let [Formula: see text] be its reciprocal with respect to the Dirichlet convolution. We study the asymptotic behavior of [Formula: see text] with regard to the asymptotic behavior of [Formula: see text] assuming that the latter one grows or decays with at most polynomial...

The anti-maximum principle for the homogeneous Dirichlet problem to −Δpu=λ|u|p−2u+f(x) with positive f∈L∞(Ω) states the existence of a critical value λf>λ1 such that any solution of this problem with λ∈(λ1,λf) is strictly negative. In this paper, we give a variational upper bound for λf and study its properties. As an important supplementary result...

Assume that a family of domain-dependent functionals $E_{\Omega_t}$ possesses a corresponding family of least energy critical points $u_t$ which can be found as (possibly nonunique) minimizers of $E_{\Omega_t}$ over the associated Nehari manifold $\mathcal{N}(\Omega_t)$. We obtain a formula for the second-order derivative of $E_{\Omega_t}$ with res...

We investigate the properties of the Cheeger sets of rotationally invariant, bounded domains $\Omega \subset \mathbb{R}^n$. For a rotationally invariant Cheeger set $C$, the free boundary $\partial C \cap \Omega$ consists of pieces of Delaunay surfaces, which are rotationally invariant surfaces of constant mean curvature. We show that if $\Omega$ i...

Let $f(n)$ be an arithmetic function with $f(1)\neq0$ and let $f^{-1}(n)$ be its reciprocal with respect to the Dirichlet convolution. We study the asymptotic behaviour of $|f^{-1}(n)|$ with regard to the asymptotic behaviour of $|f(n)|$ assuming that the latter one grows or decays with at most polynomial or exponential speed. As a by-product, we o...

We show that the elliptic equation with a non-Lipschitz right-hand side, −Δu=λ|u| β−1 u−|u| α−1 u with λ>0 and 0<α<β<1, considered on a smooth star-shaped domain Ω subject to zero Dirichlet boundary conditions, might possess a nonnegative ground state solution which violates Hopf's maximum principle only on a nonempty subset Γ of the boundary ∂Ω su...

We consider the Dirichlet problem for the nonhomogeneous equation \(-\Delta _p u -\Delta _q u = \alpha |u|^{p-2}u + \beta |u|^{q-2}u + f(x)\) in a bounded domain, where \(p \ne q\), and \(\alpha , \beta \in \mathbb {R}\) are parameters. We explore assumptions on \(\alpha \) and \(\beta \) that guarantee the resolvability of the considered problem....

We show that the elliptic equation with non-Lipschitz right-hand side, $-\Delta u = \lambda |u|^{\beta-1}u - |u|^{\alpha-1}u$ with $\lambda>0$ and $0<\alpha<\beta<1$, considered on a smooth star-shaped domain $\Omega$ subject to zero Dirichlet boundary conditions, might possess a nonnegative ground state solution which violates Hopf's maximum princ...

Assume that a family of domain-dependent functionals $E_{\Omega_t}$ possesses a corresponding family of least energy critical points $u_t$ which can be found as (possibly nonunique) minimizers of $E_{\Omega_t}$ over the associated Nehari manifold. We obtain a formula for the second-order shape derivative of $E_{\Omega_t}$ along Nehari manifold traj...

We consider the Dirichlet problem for the nonhomogeneous equation $-\Delta_p u -\Delta_q u = \alpha |u|^{p-2}u + \beta |u|^{q-2}u + f(x)$ in a bounded domain, where $p \neq q$, and $\alpha, \beta \in \mathbb{R}$ are parameters. We explore assumptions on $\alpha$ and $\beta$ that guarantee the resolvability of the considered problem. Moreover, we in...

The anti-maximum principle for the homogeneous Dirichlet problem to $-\Delta_p = \lambda |u|^{p-2}u + f(x)$ with positive $f \in L^\infty(\Omega)$ states the existence of a critical value $\lambda_f > \lambda_1$ such that any solution of this problem with $\lambda \in (\lambda_1, \lambda_f)$ is strictly negative. In this paper, we give a variationa...

Let $a_{\nu,k}$ be the $k$-th positive zero of the cross-product of Bessel functions $J_\nu(R z) Y_\nu(z) - J_\nu(z) Y_\nu(R z)$, where $\nu\geq 0$ and $R>1$. We derive an initial value problem for a first order differential equation whose solution $\alpha(x)$ characterizes the limit behavior of $a_{\nu,k}$ in the following sense: $$ \lim_{k \to \i...

We investigate strong and weak versions of maximum and comparison principles for a class of quasilinear parabolic equations with the $p$-Laplacian $$ \partial_t u - \Delta_p u = \lambda |u|^{p-2} u + f(x,t) $$ under zero boundary and nonnegative initial conditions on a bounded cylindrical domain $\Omega \times (0, T)$, $\lambda \in \mathbb{R}$, and...

We provide an explicit expression for the Pleijel constant for the planar disk and some of its sectors, as well as for $N$-dimensional rectangles. In particular, the Pleijel constant for the disk is equal to 0.4613019... Also, we characterize the Pleijel constant for some rings and annular sectors in terms of asymptotic behavior of zeros of certain...

We study the solvability of the Zakharov equation $$\Delta^2 u + (\kappa-\omega^2)\Delta u - \kappa \,\text{div} \left(e^{-|\nabla u|^2} \nabla u\right) = 0$$ in a bounded domain under homogeneous Dirichlet or Navier boundary conditions. This problem is a consequence of the system of equations derived by Zakharov to model the Langmuir collapse in p...

In this note we prove the Payne-type conjecture about the behaviour of the nodal set of least energy sign-changing solutions for the equation $-\Delta_p u = f(u)$ in bounded Steiner symmetric domains $\Omega \subset \mathbb{R}^N$ under the zero Dirichlet boundary conditions. The nonlinearity $f$ is assumed to be either superlinear or resonant. In t...

We develop the notion of higher Cheeger constants for a measurable set $\Omega \subset \mathbb{R}^N$. By the $k$-th Cheeger constant we mean the value \[h_k(\Omega) = \inf \max \{h_1(E_1), \dots, h_1(E_k)\},\] where the infimum is taken over all $k$-tuples of mutually disjoint subsets of $\Omega$, and $h_1(E_i)$ is the classical Cheeger constant of...

We study in detail the existence, nonexistence and behavior of global minimizers, ground states and corresponding energy levels of the $(p,q)$-Laplace equation $-\Delta_p u -\Delta_q u = \alpha |u|^{p-2}u + \beta |u|^{q-2}u$ in a bounded domain $\Omega \subset \mathbb{R}^N$ under zero Dirichlet boundary condition, where $p > q > 1$ and $\alpha, \be...

We investigate multiplicity and symmetry properties of higher eigenvalues and eigenfunctions of the $p$-Laplacian under homogeneous Dirichlet boundary conditions on certain symmetric domains $\Omega \subset \mathbb{R}^N$. By means of topological arguments, we show how symmetries of $\Omega$ help to construct subsets of $W_0^{1,p}(\Omega)$ with suit...

We study the dependence of least nontrivial critical levels of the energy functional corresponding to the zero Dirichlet problem $-\Delta_p u = f(u)$ in a bounded domain $\Omega \subset \mathbb{R}^N$ upon domain perturbations. The nonlinearity $f$ is assumed to be superlinear and subcritical. We show that among all (generally eccentric) spherical a...

Let $B_1$ be a ball in $\mathbb{R}^N$ centred at the origin and $B_0$ be a smaller ball compactly contained in $B_1$. For $p\in(1, \infty)$, using the shape derivative method, we show that the first eigenvalue of the $p$-Laplacian in annulus $B_1\setminus \overline{B_0}$ strictly decreases as the inner ball moves towards the boundary of the outer b...

We investigate the existence of nodal (sign-changing) solutions to the Dirichlet problem for a two-parametric family of partially homogeneous (p,q){(p,q)}-Laplace equations -Δpu-Δqu=α|u|p-2u+β|u|q-2u{-\Delta_{p}u-\Delta_{q}u=\alpha\lvert u\rvert^{p-2}u+\beta\lvert u\rvert^{q-2%
}u} where p≠q{p\neq q}. By virtue of the Nehari manifolds, the li...

We discuss several properties of eigenvalues and eigenfunctions of the $p$-Laplacian on a ball subject to zero Dirichlet boundary conditions. Among main results, in two dimensions, we show the existence of nonradial eigenfunctions which correspond to the radial eigenvalues. Also we prove the existence of eigenfunctions whose shape of the nodal set...

We construct a positive solution to a quasilinear parabolic problem
in a bounded spatial domain with the p-Laplacian and a nonsmooth
reaction function. We obtain nonuniqueness for zero initial data.
Our method is based on sub- and supersolutions and
the weak comparison principle.
Using the method of sub- and supersolutions
we construct a positive s...

We study the existence and non-existence of positive solutions for the
$(p,q)$-Laplace equation $-\Delta_p u -\Delta_q u = \alpha |u|^{p-2} u + \beta
|u|^{q-2} u$, where $p \neq q$, under the zero Dirichlet boundary condition in
$\Omega$. The main result of our research is the construction of a continuous
curve in $(\alpha,\beta)$ plane, which beco...

We prove the existence of a nodal solution with two nodal domains for the Dirichlet problem with indefinite nonlinearity
\begin{equation*}
-\Delta_p u = \lambda |u|^{p-2} u + f(x) |u|^{\gamma-2} u
\end{equation*}
in a bounded domain $\Omega \subset \mathbb{R}^n$, provided $\lambda \in (-\infty, \lambda^*_1)$, where $\lambda^*_1$ is a critical spect...

The paper is devoted to the study of two-parametric families of Dirichlet
problems for systems of equations with $p, q$-Laplacians and indefinite
nonlinearities. Continuous and monotone curves $\Gamma_f$ and $\Gamma_e$ on the
parametric plane $\lambda \times \mu$, which are the lower and upper bounds for
a maximal domain of existence of weak positi...

We study the existence of nodal solutions of a parametrized family of Dirichlet boundary value problems for elliptic equations with convex-concave nonlinearities. In the main result, we prove the existence of nodal solutions u
λ for λ ∈ (−∞, λ*0). The critical value λ*0 >0 is found by a spectral analysis procedure according to Pokhozhaev’s fibering...

We consider the behaviour of solutions to a system of homogeneous
equations with indefinite nonlinearity depending on two parameters
$(\lambda, \mu)$. Using spectral analysis a critical point
$(\lambda^*, \mu^*)$ of the Nehari manifolds and fibering methods
is introduced. We study a branch of a ground state and its
asymptotic behaviour, including t...

In a bounded connected domain Ω⊂R^N, N > 1, with a smooth boundary, we consider the Dirichlet boundary value problem for elliptic equation with a convex-concave nonlinearity
−∆u = λ|u|^{q−2} u + |u|^{γ−2} u, x ∈ Ω
u|_{∂Ω} = 0,
where 1 < q < 2 < γ < 2*. As a main result, we prove the existence of a nodal solution to this equation on the nonlocal int...