We study the nonlinear Schr\"odinger equation with $\delta'_s$ coupling of intensity $\beta\in\mathbb{R}\setminus\{0\}$ on star graph $\Gamma$ consisting $N$ half-lines. The nonlinearity has form $g(u)=|u|^{p-1}u, p>1.$ In first part paper, under certain restriction $\beta$, we prove existence ground state solution as a minimizer action functional $S_\omega$ Nehari manifold. It appears that fam...

and Applied Analysis 3 Here, S t denotes the corresponding semigroup on H, generated by problem 1.1 , and ·, · 2 is the inner product in L2 Ω . The following energy equation holds: E0 E t ∫ t 0 δ‖v τ ‖λdτ, 2.3 where E t ≡ E u t , v t ≡ 1 2 ‖v t ‖2 J u t , 2.4 J u ≡ 1 2 a u 1 2 ( γ 1 )c u − 1 r b u , 2.5 with a u ≡ ‖u‖B, b u ≡ μ‖u‖r , c u ≡ β‖∇u‖ γ 1 2 . 2.6 Here, E0 ≡ E u0, v0 is the initial en...

‎By considering a degenerate $p(x)-$Laplacian equation‎, ‎a generalized compact embedding in weighted variable‎ ‎exponent Sobolev space is presented‎. ‎Multiplicity of positive solutions are discussed by applying fibering map approach for the‎ ‎corresponding Nehari manifold‎. 

The classical scalar Nevanlinna-Pick interpolation problem has a long and distinguished history, appearing in a variety of applications in mathematics and electrical engineering. There is a vast literature on this problem and on its various far reaching generalizations; for a quick historical survey see [1] and [38]. It is widely known that the now classical algorithm for solving this problem p...

In recent works we considered an asymptotic problem for orthogonal polynomials when a Szegö measure on the unit circumference is perturbed by an arbitrary Blaschke sequence of point masses outside the unit disk. In the current work we consider a similar problem in the scattering setting. The goal of this work is to consider a new asymptotic problem; the related problems in the spectral setting ...

Underlying many of the current mathematical opportunities in digital signal processing are unsolved analog signal processing problems. For instance, digital signals for communication or sensing must map into an analog format for transmission through a physical layer. In this layer we meet a canonical example of analog signal processing: the electrical engineer’s impedance matching problem. Impe...

and Applied Analysis 3 Consequently, the condition u′ 0 0 1.11 is necessary for each solution of 1.7 . Denote usup sup{u t : t ∈ 0,∞ }. 1.12 Definition 1.2. Let u be a solution of 1.7 . If usup < L, then u is called a damped solution. If a solution u of 1.7 satisfies usup L or usup > L, then we call u a bounding homoclinic solution or an escape solution. These three types of solutions have been...

The present article serves the purpose of pursuing Geometrization of heat flow on volumetrically isothermal manifold by means of RF approach. In this article, we have analyzed the evolution of heat equation in a 3-dimensional smooth isothermal manifold bearing characteristics of Riemannian manifold and fundamental properties of thermodynamic systems. By making use of the notions of various curva...

A manifold with a smooth action of a Lie group G is called G-manifold. In this paper we consider a complete Riemannian manifold M with the action of a closed and connected Lie subgroup G of the isometries. The dimension of the orbit space is called the cohomogeneity of the action. Manifolds having actions of cohomogeneity zero are called homogeneous. A classic theorem about Riemannian manifolds...