SOME BASIC FACTS ON THE SYSTEM ∆u − Wu(u) = 0
نویسنده
چکیده
We rewrite the system ∆u − Wu(u) = 0, for u : R n → R n , in the form div T = 0, where T is an appropriate stress-energy tensor, and derive certain a priori consequences on the solutions. In particular, we point out some differences between two paradigms: the phase-transition system, with target a finite set of points, and the Ginzburg–Landau system, with target a connected manifold.
منابع مشابه
On a class of nonlinear fractional Schrödinger-Poisson systems
In this paper, we are concerned with the following fractional Schrödinger-Poisson system: (−∆s)u + V (x)u + φu = m(x)|u|q−2|u|+ f(x,u), x ∈ Ω, (−∆t)φ = u2, x ∈ Ω, u = φ = 0, x ∈ ∂Ω, where s,t ∈ (0,1], 2t + 4s > 3, 1 < q < 2 and Ω is a bounded smooth domain of R3, and f(x,u) is linearly bounded in u at infinity. Under some assumptions on m, V and f we obtain the existence of non-trivial so...
متن کاملPositive Solutions for System of Nonlinear Fractional Differential Equations in Two Dimensions with Delay
and Applied Analysis 3 Definition 2.2. For f, g ∈ E the order interval 〈f, g〉 is defined as 17 〈 f, g 〉 { h ∈ E : f ≤ h ≤ g. 2.1 Definition 2.3. A subset E ⊂ Π is called order bounded if E is contained in some order interval. Definition 2.4. A coneK is called normal if there exists a positive constant μ such that f, g ∈ V and θ ≺ f ≺ g implies that ‖f‖ ≤ μ‖g‖. We state the two fixed point resul...
متن کاملA Quantitative Comparison Theorem for Nonlinear Equations
In the present paper we establish a quantitative comparison theorem for positive solutions of the following initial value problems 8 < : (p 1 (r)(u)ju 0 j m?2 u 0) 0 + q 1 (r)f(u) = 0 u(0) = u 0 ; u 0 (0) = 0 and 8 < : (p 2 (r)(v)jv 0 j m?2 v 0) 0 + q 2 (r)f(v) = 0 v(0) = v 0 ; v 0 (0) = 0 with r > 0 and m > 1, and also show some applications of the theorem to the non-existence problem of posit...
متن کاملExistence of Some Sparse Sets of Nonstandard Natural Numbers
Answers are given to two questions concerning the existence of some sparse subsets of H = f0; 1; : : : ;H 1g N, where H is a hyper nite integer. In x1, we answer a question of Kanovei by showing that for a given cut U in H, there exists a countably determined set X H which contains exactly one element in each U -monad, if and only if U = a N for some a 2 H r f0g. In x2, we deal with a question ...
متن کاملRadial Dunkl Processes Associated with Dihedral Systems
We give some interest in radial Dunkl processes associated with dihedral systems. We write down the semi group density and as a by-product the generalized Bessel function and the W -invariant generalized Hermite polynomials. Then, a skew product decomposition, involving only independent Bessel processes, is given and the tail distribution of the first hitting time of boundary of the Weyl chambe...
متن کامل