On the Structure of the Solution Set of u″ = f(t,u), u(0) = u(1) = 0

The solution set of a Dirichlet problem x″ = f(t, x), x(0) = x(1) = 0, on a Banach space E and with f satisfying a Lipschitz condition, is homeomorphic to a closed subset of E. We prove that to an closed subset C of E there is a function f with Lipschitz constant arbitrarily close to π², such that the solution set of the corresponding Dirichlet problem is homeomorphic to C.     

Four positive solutions for the semilinear elliptic equation: -Delta u+u=a(x)u^p+f(x) in {mathbb R}^N

We consider the existence of positive solutions of the following semilinear elliptic problem in : where , , , and . Under the conditions: 1° for all , 2° as , 3° there exist and such that 4°, we show that (*) has at least four positive solutions for sufficiently small but . Received December 11, 1998 / Accepted July 16, 1999 / Published online April 6, 2000     

Metastable patterns in solutions of ut = ϵ2uxx − f(u)

We consider the equation in question on the interval 0 ≦ x ≦ 1 having Neumann boundary conditions, with f(u) = F(u), where F is a double well energy density with equal minima at u = ±1. The only stable states of the system are patternless constant solutions. But given two-phase initial data, a pattern of interfacial layers typically forms far out of equilibrium. The ensuing nonlinear relaxation process is extremely slow: patterns persist for exponentially long times proportional to exp{A_±l/?, where A = F(±1) and l is the minimum distance between layers. Physically, a tiny potential jump across a layer drives its motion. We prove the existence and persistence of these metastable patterns, and characterise accurately the equations governing their motion. The point of view is reminiscent of center manifold theory: a manifold parametrising slowly evolving states is introduced, a neighbourhood is shown to be normally attracting, and the parallel flow is characterised to high relative accuracy. Proofs involve a detailed study of the Dirichlet problem, spectral gap analysis, and energy estimates.     

Existence of multiple positive solutions for −Δu−μ(u/∣x∣2)=u2*−1+σf(x)

In this paper, we consider the semilinear elliptic problem where Ω??^N (N?3) is a bounded smooth domain such that 0∈Ω, σ>0 is a real parameter, and f(x) is some given function in L^∞(Ω) such that f(x)?0, f(x)?0 in Ω. Some existence results of multiple solutions have been obtained by implicit function theorem, monotone iteration method and Mountain Pass Lemma. Copyright © 2002 John Wiley & Sons, Ltd.     

Periodic solutions of u'(t) = −f(u(t − 1)): Multiplicity results and small period solutions

In this paper we are concerned with the differential delay equation For odd and continuous nonlinearities f: R → R . We want to prove existence and multiplicity criteria for so called “slowly oscillating” periodic solutions of (A). The oddness condition is of course a rather restrictive one, but due to results of Nussbaum [9], [10] and Peters [13],[14] and numerical studies of Jürgens, Peitgen and Saupe [7] and Hadeler [6] we know that even for odd nonlinearities f (A) may display a complicated dynamical behaviour. On the other hand the oddness condition allows a classification of symmetry properties of periodic solutions of (A).     

On the radial distribution of Julia sets of entire solutions of f(n)+A(z)f=0

This paper is devoted to studying the dynamical properties of solutions of $f^{(n)}+A(z)f=0$, where $n(?2)$ is an integer, and $A(z)$ is a transcendental entire function of finite order. We find the lower bound on the radial distribution of Julia sets of $E(z)$ provided that $E=f_1{f}_2?f_n$ and $\{f_1,f_2,\dots ,f_n\}$ is a solution base of such equations.     

Solutions of the Differential Equation u‴+λ2zu+(α−1)λ2u=0

This paper deals with the solutions of the differential equation u?+λ²zu+(α?1)λ²u=0, in which λ is a complex parameter of large absolute value and α is an arbitrary constant, real or complex. After a discussion of the structure of the solutions of the differential equation, an integral representation of the solution is given, from which the series solutions and their asymptotic representations are derived. A third independent solution is needed for the special case when α?1 is a positive integer, and two derivations for this are given. Finally, a comparison is made with the results obtained by R. E. Langer.     

Uniqueness and nonuniqueness of nodal radial solutions of sublinear elliptic equations in a ball

The following Dirichlet problem

(1.1)

is considered, where , N≥2, KC²[0,1] and K(r)>0 for 0≤r≤1, , sf(s)>0 for s≠0. Assume moreover that f satisfies the following sublinear condition: f(s)/s>f^′(s) for s≠0. A sufficient condition is derived for the uniqueness of radial solutions of (1.1) possessing exactly k−1 nodes, where . It is also shown that there exists KC^∞[0,1] such that (1.1) has three radial solutions having exactly one node in the case N=3.