Band 7, Heft 1

We consider the following elliptic system: where Ω ⊂ ℝ N , N ≥ 3 is a smooth bounded domain. If h(x) ≡ k(x) ≡ 0, the system presents a natural ℤ 2 symmetry, which guarantees the existence of infinitely many solutions. In this paper we show that the multiplicity structure can be maintained if (p,q) lies below a suitable curve in ℝ 2 .

In this paper we study the Dirichlet problem for a class of nonlinear elliptic equations in the form A(u) = H(x, u, Du), where the principal term is a Leray-Lions operator defined on W 0 1,p (Ω), and H(x, u, Du) grows with respect to Du at most like |Du| q , p-1 ≤ q ≤ p. Comparison results are obtained between the rearrangement of a solution u of Dirichlet problem quoted above and the rearrangement of the solution of a problem whose data are radially symmetric.

We give some results about the dynamics of a particle moving in Euclidean three-space under the influence of the gravitational force induced by a fixed homogeneous circle. Our main results concern (1) singularities and (2) the dynamics in the plane that contains the circle. The study presented here is purely analytic.

The aim of this paper is to study the existence and concentration of positive solutions for the coupled nonlinear Schrödinger system where ε is a small positive number, N ≥ 3, p, q > 1 satisfy and W(x), Q(x), K(x) are continuous and bounded positive functions defined in ℝ N . Combining the dual variational methods with mountain pass theorem, we prove under some conditions on W, Q, K the existence of a family of positive solutions concentrating at a point where related functionals achieve their minimum energy.

In this paper we study resonance for nonlinear oscillators from a topological point of view. The Poincaré map associated to a linear second order equation at resonance is a translation of the plane. Our aim is to prove that in the case of nonlinear equations the Poincaré map is topologically conjugate to a translation. The proof is based on a characterization of translations in terms of Lyapunov functions.

We consider the question of existence of nonnegative solutions of a class of p-Laplacian elliptic problems in the whole space ℝ N having a sign-changing nonlinearity of the form a(x)g(u).

In this paper, we establish an index theory for symplectic paths starting from the identity with a Lagrangian boundary condition. We show that in some sense the index is the intersection number of a Lagrangian path generated by the symplectic path and a constant Lagrangian path. The various properties of this index theory are developed.