Volume 3, Issue 4

We study the equation -Δu + |x| a u = |x| b |u| p-2 u with Dirichlet boundary conditions on the unit ball B(0,1) and on annuli D(R, d). The aim of this paper is to prove -under some asymptotic conditions on a, b, R and d- that there exist many nonequivalent nonradial solutions for these two problems.

We establish Hölder continuity of weak solutions to degenerate critical elliptic equations of Caffarelli-Kohn-Nirenberg type.

We consider the equation -uʺ = g(u), u(x) ∈ H 1 (ℝ). (0.1) Under general assumptions on the nonlinearity g we prove that the, unique up to translation, solution of (0.1) is at the mountain pass level of the associated functional. This result extends a corresponding result for least energy solutions when (0.1) is set on ℝ N .

This paper is devoted to the prescribed scalar curvature problem on 3 and 4- dimensional Riemannian manifolds. We give a new class of functionals which can be realized as scalar curvature. Our proof uses topological arguments and the tools of the theory of the critical points at infinity.

The aim of this paper is t o study the geodesic connectedness of a complete static Lorentzian manifold (M.〈·, ·〉 L ) such that. if K is its irrotational timelike Killing vector field, the growth of β = -〈K, K〉 L is (at most) quadratic. This is shown to be equivalent to a critical variational problem on a Riemannian manifold, where some geometrical interpretations yield the optimal results. Multiplicity of connecting geodesics is studied, especially in the timelike case, where the restrictions for the number of causal connecting geodesics are stressed. Extensions to the non-complete case, including discussions on pseudoconvexity, are also given.

In this paper, we study possible shapes of 4-body non-collinear relative equilibria for any positive masses, and give estimates on their geometric quantities.

We prove the existence of radial solutions of the quasilinear elliptic equation div(A(|Du|)Du) + f(u) = 0 in ℝ n , n > 1, where f is either negative or positive for small u > 0, possibly singular at u = 0, and grows subcritically for large u. Our proofs use only elementary arguments based on a variational identity. No differentiability assumptions are made on f.