arXiv subjects

Vladimir Bobkov

Publications and source records attributed to Vladimir Bobkov.

49 records · Page 3Linked to original sources

arXiv2019

On a property of the nodal set of least energy sign-changing solutions for quasilinear elliptic equations

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 $-Δ_p u = f(u)$ in bounded Steiner symmetric domains $Ω\subset \mathbb{R}^N$ under the zero Dirichlet boundary conditions. The nonlinearity $f$ is assumed to be either superlinear or resonant. In the latter case, least energy sign-changing solutions are second eigenfunctions of the zero Dirichlet $p$-Laplacian in $Ω$. We show that the nodal set of any least energy sign-changing solution intersects the boundary of $Ω$. The proof is based on a moving polarization argument.

math.AP↗

arXiv2018

Asymptotic relation for zeros of cross-product of Bessel functions and applications

Let $a_{ν,k}$ be the $k$-th positive zero of the cross-product of Bessel functions $J_ν(R z) Y_ν(z) - J_ν(z) Y_ν(R z)$, where $ν\geq 0$ and $R>1$. We derive an initial value problem for a first order differential equation whose solution $α(x)$ characterizes the limit behavior of $a_{ν,k}$ in the following sense: $$ \lim_{k \to \infty} \frac{a_{kx,k}}{k} = α(x), \quad x \geq 0. $$ Moreover, we show that $$ a_{ν,k} < \frac{πk}{R-1} + \frac{πν}{2R}. $$ We use $α(x)$ to obtain an explicit expression of the Pleijel constant for planar annuli and compute some of its values.

math.CA↗

arXiv2018

On maximum and comparison principles for parabolic problems with the $p$-Laplacian

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

math.AP↗

arXiv2018

On exact Pleijel's constant for some domains

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 cross-products of Bessel functions.

math.AP↗

arXiv2018

On full Zakharov equation and its approximations

We study the solvability of the Zakharov equation $$Δ^2 u + (κ-ω^2)Δu - κ\,\text{div} \left(e^{-|\nabla u|^2} \nabla u\right) = 0$$ 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 plasma physics. Assumptions for the existence and nonexistence of a ground state solution as well as the multiplicity of solutions are discussed. Moreover, we consider formal approximations of the Zakharov equation obtained by the Taylor expansion of the exponential term. We illustrate that the existence and nonexistence results are substantially different from the corresponding results for the original problem.

math.AP↗

arXiv2017

On the higher Cheeger problem

We develop the notion of higher Cheeger constants for a measurable set $Ω\subset \mathbb{R}^N$. By the $k$-th Cheeger constant we mean the value \[h_k(Ω) = \inf \max \{h_1(E_1), \dots, h_1(E_k)\},\] where the infimum is taken over all $k$-tuples of mutually disjoint subsets of $Ω$, and $h_1(E_i)$ is the classical Cheeger constant of $E_i$. We prove the existence of minimizers satisfying additional "adjustment" conditions and study their properties. A relation between $h_k(Ω)$ and spectral minimal $k$-partitions of $Ω$ associated with the first eigenvalues of the $p$-Laplacian under homogeneous Dirichlet boundary conditions is stated. The results are applied to determine the second Cheeger constant of some planar domains.

math.AP↗

arXiv2017

Remarks on minimizers for $(p,q)$-Laplace equations with two parameters

We study in detail the existence, nonexistence and behavior of global minimizers, ground states and corresponding energy levels of the $(p,q)$-Laplace equation $-Δ_p u -Δ_q u = α|u|^{p-2}u + β|u|^{q-2}u$ in a bounded domain $Ω\subset \mathbb{R}^N$ under zero Dirichlet boundary condition, where $p > q > 1$ and $α, β\in \mathbb{R}$. A curve on the $(α,β)$-plane which allocates a set of the existence of ground states and the multiplicity of positive solutions is constructed. Additionally, we show that eigenfunctions of the $p$- and $q$-Laplacians under zero Dirichlet boundary condition are linearly independent.

math.AP↗

arXiv2017

On multiplicity of eigenvalues and symmetry of eigenfunctions of the $p$-Laplacian

We investigate multiplicity and symmetry properties of higher eigenvalues and eigenfunctions of the $p$-Laplacian under homogeneous Dirichlet boundary conditions on certain symmetric domains $Ω\subset \mathbb{R}^N$. By means of topological arguments, we show how symmetries of $Ω$ help to construct subsets of $W_0^{1,p}(Ω)$ with suitably high Krasnosel'ski\uı genus. In particular, if $Ω$ is a ball $B \subset \mathbb{R}^N$, we obtain the following chain of inequalities: $$ λ_2(p;B) \leq \dots \leq λ_{N+1}(p;B) \leq λ_\ominus(p;B). $$ Here $λ_i(p;B)$ are variational eigenvalues of the $p$-Laplacian on $B$, and $λ_\ominus(p;B)$ is the eigenvalue which has an associated eigenfunction whose nodal set is an equatorial section of $B$. If $λ_2(p;B)=λ_\ominus(p;B)$, as it holds true for $p=2$, the result implies that the multiplicity of the second eigenvalue is at least $N$. In the case $N=2$, we can deduce that any third eigenfunction of the $p$-Laplacian on a disc is nonradial. The case of other symmetric domains and the limit cases $p=1$, $p=\infty$ are also considered.

math.AP↗

arXiv2017

On some unexpected properties of radial and symmetric eigenvalues and eigenfunctions of the $p$-Laplacian on a disk

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 cannot occur in the linear case $p=2$. Moreover, the limit behavior of some eigenvalues as $p \to 1+$ and $p \to +\infty$ is studied.

math.AP↗

arXiv2017

On sign-changing solutions for $(p,q)$-Laplace equations with two parameters

We investigate the existence of nodal (sign-changing) solutions to the Dirichlet problem for two-parametric family of partially homogeneous $(p,q)$-Laplace equations $-Δ_p u -Δ_q u=α|u|^{p-2}u+β|u|^{q-2}u$ where $p \neq q$. By virtue of the Nehari manifolds, linking theorem, and descending flow, we explicitly characterize subsets of $(α,β)$-plane which correspond to the existence of nodal solutions. In each subset the obtained solutions have prescribed signs of energy and, in some cases, exactly two nodal domains. The nonexistence of nodal solutions is also studied. Additionally, we explore several relations between eigenvalues and eigenfunctions of the $p$- and $q$-Laplacians in one dimension.

math.AP↗

arXiv2016

On the strict monotonicity of the first eigenvalue of the $p$-Laplacian on annuli

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 ball. The analogous results for the limit cases as $p \to 1$ and $p \to \infty$ are also discussed. Using our main result, further we prove the nonradiality of the eigenfunctions associated with the points on the first nontrivial curve of the Fučik spectrum of the $p$-Laplacian on bounded radial domains.

math.AP↗

arXiv2015

Maximal existence domains of positive solutions for two-parametric systems of elliptic equations

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 $Γ_f$ and $Γ_e$ on the parametric plane $λ\times μ$, which are the lower and upper bounds for a maximal domain of existence of weak positive solutions are introduced. The curve $Γ_f$ is obtained by developing our previous work \cite{BobkovIlyasov} and it determines a maximal domain of the applicability of the Nehari manifold and fibering methods. The curve $Γ_e$ is derived explicitly via minimax variational principle of the extended functional method.

math.AP↗

arXiv2014

On positive solutions for $(p,q)$-Laplace equations with two parameters

We study the existence and non-existence of positive solutions for the $(p,q)$-Laplace equation $-Δ_p u -Δ_q u = α|u|^{p-2} u + β|u|^{q-2} u$, where $p \neq q$, under the zero Dirichlet boundary condition in $Ω$. The main result of our research is the construction of a continuous curve in $(α,β)$ plane, which becomes a threshold between the existence and non-existence of positive solutions. Furthermore, we provide the example of domains $Ω$ for which the corresponding first Dirichlet eigenvalue of $-Δ_p$ is not monotone w.r.t. $p > 1$.

math.AP↗