Band 14, Heft 3

In this paper, let Σ ⊂ R 2n be a symmetric compact convex hypersurface. We prove that either there are infinitely many closed characteristics or there exist at least 2[n/2] nonhyperbolic closed characteristics on Σ. Due to the example of weakly non-resonant ellipsoids, this estimate is sharp when n is even. Moreover, we prove that if Σ carries exactly n closed characteristics, then at least n − 2 of them possess irrational mean indices.

In this paper we consider the following nonlinear parabolic problem where Ω ⊂ ℝ N , N > 2, is a bounded domain with 0 ∈ Ω and 1 < p < 3. The main goal of this work is to analyze the influence of the gradient term in order to obtain the existence of a positive solution for the largest class of data ( f, u 0 ) and for all λ > 0.

We study a nonlinear parametric elliptic equation (nonlinear eigenvalue problem) driven by a nonhomogeneous differential operator. Our setting incorporates equations driven by the p-Laplacian, the (p, q)-Laplacian, and the generalized p-mean curvature differential operator. Applying variational methods we show that for λ > 0 (the parameter) sufficiently large the problem has at least three nontrivial smooth solutions whereby one is positive, one is negative and the last one has changing sign (nodal). In the particular case of (p, 2)-equations, using Morse theory, we produce another nodal solution for a total of four nontrivial smooth solutions.

We prove an existence result for a fourth order elliptic equation where the associated functional does not satisfy the Palais-Smale condition. We use some truncations argument and L ∞ -norm estimates.

In this paper we build Non-singular Morse-Smale flows on S 3 with unknotted and unlinked saddle orbits by identifying fat round handles along their boundaries. This way of building the flows enables to get their phase portraits. We also show that the presence of heteroclinic trajectories imposes an order in the round handle decomposition of these flows; this order is total for NMS flows composed of one repulsive, one attractive and n unknotted saddle orbits, for n ≥ 1.

This article concerns a class of nonlocal fractional Laplacian problems depending of three real parameters. More precisely, by using an appropriate analytical context on fractional Sobolev spaces due to Servadei and Valdinoci (in order to correctly encode the Dirichlet boundary datum in the variational formulation of our problem) we establish the existence of three weak solutions for fractional equations via a recent abstract critical point result for differentiable and parametric functionals recently proved by Ricceri.

In this paper we study multi-valued parabolic variational inequalities involving quasilinear parabolic operators, and multi-valued integral terms over the underlying parabolic cylinder as well as over parts of the lateral parabolic boundary, where the multi-valued functions involved are assumed to be upper semicontinuous only. Note, since lower semicontinuous multi-valued functions allow always for a Carathéodory selection, this case can be considered as the trivial case, and therefore will be omitted. Our main goal is threefold: First, we provide an analytical frame work and an existence theory for the problems under consideration. Unlike in recent publications on multi-valued parabolic variational inequalities, the closed convex set K representing the constraints is not required to possess a nonempty interior. Second, we prove enclosure and comparison results based on a recently developed sub-supersolution method due to the authors. Third, we consider classes of relevant generalized parabolic variational-hemivariational inequalities that will be shown to be special cases of the multi-valued parabolic variational inequalities under consideration.

Consider in a real Hilbert space H the differential equation (inclusion) (E): p(t)u″(t) + q(t)u′(t) ∈ Au(t) + f (t) for a.a. t ∈ ℝ + = [0,∞), subject to the condition , where A: D(A) ⊂ H → H is a (possibly set-valued) maximal monotone operator whose range contains 0; p, q ∈ L ∞ (ℝ + ), with ess inf p > 0 and either ess inf q > 0 or ess sup q < 0. Eq. (E) has been investigated under various assumptions by Barbu [1]-[2], Brezis [4], Véron [9]-[10] and others. Recently (see Moroşanu [8]), we proved existence and uniqueness of a strong solution to Eq. (E) subject to u(0) = x ∈ D(A) in the weighted space X = L b 2 (ℝ + ; H), where b(t) = a(t)/p(t), a(t) = exp(∫ 0 t q(s)/p(s) ds), under our weak conditions on p and q (see above). Here we extend this result to the general case , while keeping the other assumptions unchanged, thus solving a long standing problem. In addition, the proof of our recent result, which is the starting point of the present paper, is considerably simplified. Furthermore, some qualitative properties of solutions are pointed out, an application to a minimization problem is given, and some open problems are formulated.

We deal with the existence of nonzero solution for the quasilinear Schrödinger equation −Δu + V(x)u − Δ(u 2 )u = g(x, u), x ∈ ℝ N , u ∈ H 1 (ℝ N ), where V is a positive potential and the nonlinearity g(x, s) behaves like K 0 (x)s at the origin and like K ∞ (x)|s| p , 1 ≤ p ≤ 3, at infinity. In the proofs we apply minimization methods.

We consider the Zakharov system with quantum effects in the framework of the scalar mode in two space dimensions iE t + ΔE = nE + ΓΔ 2 E, n t = −∇ · V, (ZSQ) V t = −∇n − ∇|E| 2 + Γ∇Δn, which is related to quantum corrections to the Zakharov system for Langmuir waves in plasma. We prove the existence and uniqueness of global smooth solutions to the Cauchy problem for (ZSQ) in the Sobolev space through making a priori integral esimates and utilizing the so-called continuity method.

In this paper the method of moving spheres is used to derive a Harnack-type inequality for positive solutions of where n ≥ 4, ℝ + n is the upper half of Euclidean space and B 1 + is the upper half unit ball. Under suitable assumptions on K(x) and c(x), we show that there is a positive constant C such that for all positive solutions u, a Harnack-type inequality holds. As a consequence of this inequality we obtain the following energy estimate .

We consider a class of nonlinear stationary Schrödinger-type equations and we are concerned with sufficient properties that guarantee the existence of multiple solutions in a suitable Sobolev space with variable exponents. We first establish that in the case of small perturbations, the problem admits at least two weak solutions. Next, in the case of convexconcave nonlinearities, we obtain conditions for the existence of infinitely many solutions with high (resp., small) energies. The arguments combine variational techniques with a careful analysis of the energy levels.

We examine the following fourth order Hénon equation Δ 2 u = |x| α u p in ℝ N , where 0 < α. Define the Hardy-Sobolev exponent . We show that in dimension N = 5 there are no positive bounded classical solutions provided 1 < p < p4(α).

Finite time collapse of solutions to the generalized three-dimensional nonlocal nonlinear Schrödinger equations is studied. This result is achieved through establishing some a priori estimates for the nonlocal terms and by introducing a type of virial identities.

For a nonautonomous dynamics defined by a sequence of linear operators, we consider the notion of an exponential dichotomy with respect to a sequence of norms and we characterize it completely in terms of the admissibility in pairs of spaces (ℓ p , ℓ q ), with p and q not necessarily equal. This includes the notion of a nonuniform exponential dichotomy as a very special case. Moreover, we consider a general noninvertible dynamics and a strong nonuniform exponential behavior. The latter is the typical situation from the point of view of smooth ergodic theory.