Volume 12, Issue 3

We consider the problem −Δu + (V ∞ + V(x))u = |u| p−2 u, u ∈ H 0 1 (Ω), where Ω is an exterior domain in ℝ N , V ∞ > 0, V ∈ C 0 (ℝ N ), inf ℝN V > −V ∞ and V(x) → 0 as |x| → ∞. Under symmetry conditions on Ω and V, and some assumptions on the decay of V at infinity, we show that there is an effect of the topology of the orbit space of certain subsets of the domain on the number of low energy sign changing solutions to this problem.

Using a recent modified version of the Poincaré-Birkhoff fixed point theorem [19], we study the existence of one-signed T-periodic solutions and sign-changing subharmonic solutions to the second order scalar ODE u′′ + f (t, u) = 0, being f : ℝ × ℝ → ℝ a continuous function T-periodic in the first variable and such that f (t, 0) ≡ 0. Partial extensions of the results to a general planar Hamiltonian systems are given, as well.

We consider the nonlinear inverse bifurcation problem −u′′(t) + f (u(t)) = λu(t), u(t) > 0, t ∈ I := (0, 1), u(0) = u(1) = 0, where λ > 0 is a positive parameter, and the nonlinear term f (u) is unknown. We show that, if the asymptotic behavior of the L q -bifurcation curve λ = λ q (α) (1 ≤ q < ∞) of the positive solutions is understood well, then we are able to obtain f (u) and to characterize the asymptotic behavior of f(u) as u → ∞.

Using a topological approach we prove the existence of infinitely many periodic solutions, as well as the presence of symbolic dynamics for second order nonlinear differential equations of the form −ü − μu + g(t)h(u) = 0 where μ > 0 is a given constant and g : ℝ → ℝ is a periodic positive weight function. Our main application concerns the study of the case in which h(u) is a cubic nonlinearity. Such a choice is motivated by previous investigations dealing with the nonlinear Schrödinger equation

In a previous paper [15] we introduced the Sturm-Liouville (SL) hierarchy of evolution equations. This hierarchy includes the Korteveg-de Vries (K-dV) and the Camassa-Holm (CH) hierarchies. We also defined and discussed in detail the algebro-geometric solutions of the SL-hierarchy. In this paper, we broaden the class of algebro-geometric solutions in a substantial way. Namely, we define and discuss solutions of the SL-hierarchy lying in an isospectral class of the Sturm-Liouville problem −(pφ′)′ + qφ = λyφ, which is determined by data related to a Riemann surface of “infinite genus”.

In this paper, we apply the discrete Littlewood-Paley-Stein analysis to prove the duality theorem of weighted multi-parameter Hardy spaces associated with Zygmund dilations, i.e., (HpZ (ω))∗ = CMO Z p (ω) for 0 < p ≤ 1. Our theorems extend the H Z p - CMO Z p duality theorems established in [13] (see also [12]) for non-weighted multi-parameter Hardy spaces associated with the Zygmund dilation.

We study the existence of ground state solution, cylindrically symmetric solution, breaking cylindrical symmetry of ground state solution of the following Hardy-Sobolev type equa- tion with weighted p-Laplacian: where

A detailed study of two classes of oscillatory global (and blow-up) solutions was began in [20] for the semilinear heat equation in the subcritical Fujita range with bounded integrable initial data u(x, 0) = u 0 (x). This study is continued and extended here for the 2mth-order heat equation, for m ≥ 2, with non-monotone nonlinearity with the same initial data u 0 . The fourth order biharmonic case m = 2 is studied in greater detail. The blow-up Fujita-type result for (0.2) now reads as follows: blow-up occurs for any initial data u0 with positive first Fourier coefficient: ∫ u 0 (x) dx > 0, i.e., as for (0.1), any such arbitrarily small initial function u 0 (x) leads to blow-up. The construction of two countable families of global sign changing solutions is performed on the basis of bifurcation/branching analysis and a further analytic-numerical study. In particular, a countable sequence of bifurcation points of similarity solutions is obtained:

In this paper we study semilinear variational inequalities driven by an elliptic operator not in divergence form modeled by where Ω is a bounded domain of RN, N ≥ 3, with smooth boundary, A is the elliptic operator, not in divergence form, given by Here a ij , a i , i,j = 1,...,N , and a 0 satisfy suitable regularity conditions, while 1 < s < 4/(N − 2) and the obstacle ψ is a function sufficiently smooth. Even if this problem is not variational in nature, we will prove the existence of non-trivial non-negative solutions for it, performing a variational approach combined with a penalization technique. This kind of approach seems to be new for problems of this type. We also prove a C 1,α -regularity result for the solutions of our problem.

We discuss existence and multiplicity of positive solutions of the Dirichlet problem for the quasilinear ordinary differential equation . Depending on the behaviour of f = f (t, s) near s = 0, we prove the existence of either one, or two, or three, or infinitely many positive solutions. In general, the positivity of f is not required. All results are obtained by reduction to an equivalent non-singular problem to which variational or topological methods apply in a classical fashion.

We show the existence of standing-wave solutions to a coupled non-linear Klein-Gordon equation. Our solutions are obtained as minimizers of the energy under a two-charges constraint. We prove that the ground state is stable and that standing-waves are orbitally stable under a non-degeneracy assumption.

Let ℍ n = ℝ 2n × ℝ be the n−dimensional Heisenberg group, be its subelliptic gradient operator, and ρ (ξ) = ( |z| 4 + t 2 ) 1/4 for ξ = (z, t) ∈ ℍ n be the distance function in ℍ n . Denote ℍ = ℍ n , Q = 2n + 2 and Q′ = Q/(Q − 1). Let Ω be a bounded domain with smooth boundary in ℍ. Motivated by the Moser-Trudinger inequalities on the Heisenberg group, we study the existence of solution to a nonuniformly subelliptic equation of the form where f : Ω × ℝ → ℝ behaves like exp ( α |u| Q′ ) when |u| → ∞. In the case of Q−sub- Laplacian we will apply minimax methods to obtain multiplicity of weak solutions.