Volume 9, Issue 4

We study the action functional associated to a smooth Lagrangian function on the tangent bundle of a manifold, having quadratic growth in the velocities. We show that, although the action functional is in general not twice differentiable on the Hilbert manifold consisting of H 1 curves, it is a Lyapunov function for some smooth Morse-Smale vector field, under the generic assumption that all the critical points are non-degenerate. This fact is suffcient to associate a Morse complex to the Lagrangian action functional.

In this paper we consider the problem where Ω = ℝ N ∖ ω̅, ω being a nonempty open bounded domain of ℝ N having smooth boundary We prove that, if u 0 and f are nonnegative and satisfy suitable conditions, there exist at least two positive solutions. Moreover, the existence of a solution is shown to hold even when no condition on the sign of u 0 and f is assumed.

We study the equation (℘A) (-i▽ + A) 2 u + V u = |u| p-2 u, where A ∈C 1,α (ℝ N ,ℝ N ) and V ∈C 0,α (ℝ N ) are 2π-periodic in each variable, V > 0; and p ∈(2; 2*) with 2* := ∞ if N = 2 and We address two questions: first, the gauge-dependence problem for 2π-periodic solutions u : ℝ N → C and second, the multiplicity of Bloch solutions. Unlike the nonperiodic case where problem (℘A) is basically independent of A (it is gauge invariant), in the periodic case this is far from being true. Under some assumptions on A we show that, if there exists a one-to-one correspondence between the 2π-periodic solutions of (℘A) and those of (℘A + z) preserving their absolute value, then z lies in a subset of measure zero of ℝ N . We use this fact to show the existence of an uncountable set of Bloch solutions with real quasimomentum.

We consider weakly nonlinear elliptic equations with small diffusion in the case where the nonlinearity has a non-nodal zero. We show that there is an unexpected connection with problems with infinite boundary values.

Let X be a reflexive Banach space and f : X → ℝ a Gâteaux differentiable function with f′ demicontinuous and locally of class (S) + . We prove that each isolated critical point of f has critical groups of finite type and that the Poincaré-Hopf formula holds. We also show that quasilinear elliptic equations at critical growth are covered by this result.

In this paper we prove some new symmetry results for the extremals of the Caffarelli-Kohn-Nirenberg inequalities, in any dimension larger or equal to 2.

We consider quasilinear systems of second order elliptic equations on ℝ N . Using a continuation theorem based on the topological degree for C 1 - Fredholm maps, we derive global properties of a maximal connected set of solutions which decay exponentially to zero at in¯nity. These results are used to treat a problem concerning the equilibrium of an elastic body occupying the whole space and subjected to a one parameter family of localized external forces.

Let (M, g) be a complete Riemannian manifold, Ω ⊂ M an open subset whose closure is diffeomorphic to an annulus. If ∂Ω is smooth and it satisfies a strong concavity assumption, then it is possible to prove that there are at least two geometrically distinct geodesics in Ω̅ = Ω ∪ ∂Ω starting orthogonally to one connected component of ∂Ω and arriving orthogonally onto the other one. The results given in [6] allow to obtain a proof of the existence of two distinct homoclinic orbits for an autonomous Lagrangian system emanating from a nondegenerate maximum point of the potential energy, and a proof of the existence of two distinct brake orbits for a class of Hamiltonian systems. Under a further symmetry assumption, it is possible to show the existence of at least dim(M) pairs of geometrically distinct geodesics as above, brake orbits and homoclinics.

We present an alternative variational principle for the geodesics of a Randers metric. We define a functional I on the manifold of H 1,2 curves joining two points on a Randers manifold (M, F) defined by a Riemannian metric g and a differential form ω. The functional I is of class C 2 , so it is more regular than the usual action functional, which is only of class C 1,1 . Moreover the critical points of the functional I are the geodesics for the Randers metric F reparameterized with Riemannian length g constant. Some global properties of the functional I are investigated. Finally some applications to the Fermat principle in stationary spacetimes will be presented.

Given (M, g) the set {(ε, h) ∈ (0, 1) × S k : any solution u ∈ A of -ε 2 Δ g+h u+u = (u + ) p-1 on M is non degenerate} is an open dense subset of (0, 1) × B ρ where B ρ is a suitable ball centered at 0 with radius ρ in the Banach space S k . Here S k is the space of all C k symmetric covariant 2-tensors on M and A is a bounded subset of H g 1 (M) which does not contain the constant function 1.

Earlier papers have studied the existence of solutions of reversible Hamiltonian systems that are heteroclinic between a pair of minimizing periodic solutions. This note extends some of that work by allowing more degenerate behavior for the class of periodic minimizers.

In this paper we prove the almost everywhere convergence of the gradient of Palais-Smale sequences, allowing us to apply the Brezis-Lieb lemma. This leads us to show that infima are attained, and thus to prove the existence of optimal solutions for some critical problems. Our method does not use the concentration- compactness principle.