Search

The article studies the exponential localization of eigenfunctions associated with isolated eigenvalues of Schrödinger operators on infinite metric graphs. We strengthen the result obtained in [3] providing a bound for the rate of exponential localization in terms of the distance between the eigenvalue and the essential spectrum. In particular, if the spectrum is purely discrete, then the eigenfunctions decay super-exponentially. [ABSTRACT FROM AUTHOR]

We investigate the critical period bifurcations of the system ẋ = ix + xx¯(ax3 + bx2x¯ + x¯x¯2 + dx¯3) studied in [6]. We prove that at most three critical periods can bifurcate from any nonlinear center of the system. [ABSTRACT FROM AUTHOR]

We study the degenerate elliptic equation - div(|x| α∇u) = f(u) + tφ(x) + h(x) in a bounded open set Ω with homogeneous Neumann boundary condition, where α ∈ (0, 2) and f has a linear growth. The main result establishes the existence of real numbers t* and t* such that the problem has at least two solutions if t ≤ t*, there is at least one solution if t* < t ≤ t*, and no solution exists for all t > t*. The proof combines a priori estimates with topological degree arguments. [ABSTRACT FROM AUTHOR]

We investigate regular points on the boundaries of elliptic systems with non-standard growth, in particular, so-called Orlicz growth. A regular point on the boundary in this paper is a point for which a weak solution to a system is Hölder continuous in a neighborhood. Here, we assume that the boundary of a domain and the boundary data are C1, and that a system has VMO (vanishing mean oscillation) type coefficients. [ABSTRACT FROM AUTHOR]

Using a functional-analytic approach for two-point boundary value problems, for a large class of oceanic vorticities, we establish the existence of solutions to a model for the jet flows of the Antarctic circumpolar current with no azimuthal variations. In our approach we rely on the stereographic projection to pass from spherical to planar coordinates. [ABSTRACT FROM AUTHOR]

In this article, we study the existence of solutions for the semilinear elliptic equation Δ2u - a(x)Δu = b(x)|u|p-2u with Navier boundary condition u = Δu = 0 on ∂Ω, where Ω is a bounded domain with smooth boundary and 2 < p < 2 *. We consider two different assumptions on the potentials a and b, including the case of sign-changing weights. The approach is based on the Nehari manifold with variational arguments about the corresponding fibering map, which ensures the multiple results. [ABSTRACT FROM AUTHOR]

In this article we study spectral properties of non-local boundaryvalue problem for an equation of parabolic-hyperbolic type. The non-local condition binds the solution values at points on boundaries of the parabolic and hyperbolic parts of the domain with each other. Nonlocal boundary conditions of such type are called Frankl-type conditions. This problem was first formulated by Kal'menov and Sadybekov who proved the unique strong solvability. In this article we investigate one particular case of this problem, for which we show that the problem does not have eigenvalues. [ABSTRACT FROM AUTHOR]

In this article, using variational methods, we study the existence of solutions for the Kirchhoff-type problem involving tempered fractional derivatives M (∫R |Dα,λ+ u(t)|2dt) Dα,λ− (Dα,λ+ u(t)) = f(t, u(t)), t∈R, u∈Wα,2λ (R), where Dα,λ± u(t) are the left and right tempered fractional derivatives of order α ∈ (1/2, 1], λ > 0, Wα,2λ (R) represent the fractional Sobolev space, f ∈ C(R × R, R) and M ∈ C(R+, R+). [ABSTRACT FROM AUTHOR]

This article concerns the initial-boundary value problem for nonlinear pseudo-parabolic equation ut - uxxt - (1 + µ(ux))uxx + (1 + σ(ux))u = f(x, t), 0 < x < 1, 0 < t < T, u(0, t) = u(1, t) = 0, u(x, 0) = ũ0(x), where f, ũ0, µ, σ are given functions. Using the Faedo-Galerkin method and the compactness method, we prove that there exists a unique weak solution u such that u ∈ L∞(0, T; H10 ∩ H2), u' ∈ L2 (0, T; H10) and ‖u‖L∞(QT) ≤ max{‖u0(x)‖L∞(Ω), ‖f‖ L∞(QT)}. Also we prove that the problem has a unique global solution with H1 -norm decaying exponentially as t → +∞. Finally, we establish the existence and uniqueness of a weak solution of the problem associated with a periodic condition. [ABSTRACT FROM AUTHOR]

This article discusses the application of a fractional-like derivative of Lyapunov-type functions in the stability analysis of solutions of perturbed motion equations with a fractional-like derivative of the state vector. The main theorems of the direct Lyapunov method for this class of motion equations are established. [ABSTRACT FROM AUTHOR]

We apply some generalized maximum principles for establishing uniqueness and nonexistence results concerning maximal spacelike hypersurfaces immersed in a generalized Robertson-Walker (GRW) spacetime, which is supposed to obey the so-called timelike convergence condition (TCC). As application, we study the uniqueness and nonexistence of entire solutions of a suitable maximal spacelike hypersurface equation in GRW spacetimes obeying the TCC. [ABSTRACT FROM AUTHOR]

We prove a global Orlicz estimate for the gradient of weak solutions to a class of nonlinear obstacle problems with partially regular nonlinearities in nonsmooth domains. It is assumed that the nonlinearities are merely measurable in one spatial variable and have sufficiently small BMO semi-norm in the other variables, and the boundary of underlying domain is Reifenberg flat. [ABSTRACT FROM AUTHOR]

In this article we study quasilinear elliptic equations with a singular operator and at critical Sobolev growth. We prove the existence of positive solutions. [ABSTRACT FROM AUTHOR]

In this work, we study a class of Schrödinger-Newton systems with singular and critical growth terms in unbounded domains. By using the variational methods and the Brézis-Lieb [6] classical technique, the existence and multiplicity of positive solutions are established. [ABSTRACT FROM AUTHOR]

We study a singularly perturbed PDE with cubic nonlinearity depending on a complex perturbation parameter ∊. This is a continuation of the precedent work [22] by the first author. We construct two families of sectorial meromorphic solutions obtained as a small perturbation in ∊ of two branches of an algebraic slow curve of the equation in time scale. We show that the nonsingular part of the solutions of each family shares a common formal power series in ∊ as Gevrey asymptotic expansion which might be different one to each other, in general. [ABSTRACT FROM AUTHOR]

We prove that all entire stable W1,ploc solutions of weighted quasilinear problem − div(w(x)|∇u|p−2∇u) = f(x)|u|q−1u must be zero. The result holds true for p ≥ 2 and p − 1 < q < qc(p, N, a, b). Here b > a − p and qc(p, N, a, b) is a new critical exponent, which is infinity in low dimension and is always larger than the classic critical one, while w, f ∈ L1 loc(RN) are nonnegative functions such that w(x) ≤ C1|x|a and f(x) ≥ C2|x|b for large |x|. We also construct an example to show the sharpness of our result. [ABSTRACT FROM AUTHOR]

We consider the fractional differentiation operators in a variety of senses. We show that the strong accretive property is the common property of fractional differentiation operators. Also we prove that the sectorial property holds for operators second order with fractional derivative in lower terms, we explore the location of spectrum and resolvent sets and show that the generalized spectrum is discrete. We prove that there is two-sided estimate for eigenvalues of real component of operators second order with fractional derivative in lower terms. [ABSTRACT FROM AUTHOR]

In this article, we extend solution of inverse nodal problem for one-dimensional p-Laplacian equation to the case when the boundary condition is polynomially eigenparameter. To find the spectral data as eigenvalues and nodal parameters, a Prüfer substitution is used. Then, we give a reconstruction formula of the potential function by using nodal lengths. This method is similar to used in [24], and our results are more general. [ABSTRACT FROM AUTHOR]

In this article we discuss a method for constructing trace formulae for the heat-volume potential of the time-fractional diffusion equation to lateral surfaces of cylindrical domains and use these conditions to construct as well as to study a nonlocal initial boundary value problem for the time-fractional diffusion equation. [ABSTRACT FROM AUTHOR]

In this article we consider eigenvalue problems for fourth-order ordinary differential equation with spectral parameter in boundary conditions. We study the location of eigenvalues on the real axis, find the multiplicities of eigenvalues, investigate the oscillation properties of eigenfunctions, and the basis properties in the space Lp, 1 < p < ∞, of the subsystems of eigenfunctions of this problem. [ABSTRACT FROM AUTHOR]

We derive a priori bounds for positive supersolutions of -Δpu = ρ(x)f(u), where p > 1 and Δp is the p-Laplace operator, in a smooth bounded domain of RN with zero Dirichlet boundary conditions. We apply our results to the nonlinear elliptic eigenvalue problem -Δpu = λf(u), Dirichlet boundary condition, where f is a nondecreasing continuous differentiable function on such that f(0) > 0, f(t)¹=(p-1) is superlinear at infinity, and give sharp upper and lower bounds for the extremal parameter λ*p. In particular, we consider the nonlinearities f(u) = eu and f(u) = (1 + u)m (m > p - 1) and give explicit estimates on λ*p. As a by-product of our results, we obtain a lower bound for the principal eigenvalue of the p-Laplacian that improves obtained results in the recent literature for some range of p and N. [ABSTRACT FROM AUTHOR]

Using variational techniques we prove an eigenvalue theorem for a stationary p(x)-Kirchhoff problem, and provide an estimate for the range of such eigenvalues. We employ a specific family of test functions in variableexponent Sobolev spaces. Our approach permits to handle both non-degenerate and degenerate Kirchhoff coefficients. [ABSTRACT FROM AUTHOR]

We consider the nonlinear eigenvalue problem u"(t) + λ(u(t)p = u(t)q) = 0, u(t) > 0, -1 < t < 1, u(1) = u(-1) = 0, where 1 < p < q are constants and λ > 0 is a parameter. It is known in [13] that the bifurcation curve λ(α) consists of two branches, which are denoted by λ±(α). Here, α = ∥uλ∥∞. We establish the asymptotic behavior of the turning point αp of λ(α), namely, the point which satisfies dλ(αp)=dα = 0 as p → q and p → 1. We also establish the asymptotic formulas for λ+(α) and λ-(α) as α → 1 and α → 0, respectively. [ABSTRACT FROM AUTHOR]

We study a question on stability and instability of the basis prop- erty of a system of eigenfunctions of the Sturm - Liouville operator, with an integral perturbation of anti-periodic type on the boundary conditions. [ABSTRACT FROM AUTHOR]

We consider the semilinear eigenvalue problem -u"(t) + k(t)u(t)p = λu(t), u(t) > 0, t ∈ I := (-1/2, 1/2), u(-1/2) = u(1/2) = 0 where p > 1 is a constant, and λ > 0 is a parameter. We propose a new inverse bifurcation problem. Assume that k(t) is an unknown function. Then can we determine k(t) from the asymptotic behavior of the bifurcation curve? The purpose of this paper is to answer this question affirmatively. The key ingredient is the precise asymptotic formula for the Lq-bifurcation curve λ = λ(q, α) as α → ∞ (1 ≤ q < ∞), where α := ||k1/(p-1)uλ||q. [ABSTRACT FROM AUTHOR]