Volume 6, Issue 1

In this paper we are interested in existence of solutions for some nonlinear elliptic equations with principal part having degenerate coercivity. The model case is with Ω bounded open subset of ℝ N , N ≥ 2 and f a datum in some Lebesgue space.

In this paper we consider the existence and the compactness of Riemannian metrics of prescribed mean curvature and zero boundary mean curvature on a three dimensional manifold with umbilic boundary (M, g 0 ). We prove that for three dimensional manifolds with umbilic boundaries, which are not conformally equivalent to the three dimensional standard half sphere, any positive function can be realized as the scalar curvature of a Riemannian metric g conformal to g 0 with respect to which the boundary has zero mean curvature. Moreover, all such metrics stay bounded with respect to the C 2,α -topology and in the nondegerate case Morse inequalities hold.

This paper deals with a class of nonlinear elliptic Dirichlet boundary value problems where the combined effects of a sublinear and a superlinear term allow us to establish some existence and multiplicity results.

After establishing a relation between the Hill’s equations and the Emarkov-Pinney equations, we are able to use degree theory to remove a technical assumption in [14], and give a complete set of non-resonance conditions for differential equations with repulsive singularities.

We study the self-similar solutions of any sign of the equation u t - div(|▽u| p-2 ▽u) = |u| q-1 u, in ℝ N , where p, q > 1. We extend the results of Haraux-Weissler obtained for p = 2 to the case q > p-1 > 0. In particular we study the existence of slow or fast decaying solutions. For given t > 0, the fast solutions u(t, .) have a compact support in ℝ N when p > 2; and |x| p/(2-p) u(t, x) is bounded at infinity when p < 2: We describe the behaviour for large |x| of all the solutions. According to the position of q with respect to the first critical exponent p - 1 + p/N and the critical Sobolev exponent q*, we study the existence of positive solutions, or the number of the zeros of u(t, .). We prove that any solution u(t, .) is oscillatory when p < 2 and q is closed to 1.

We show the existence of periodic solutions of certain singularly perturbed systems having symmetry properties. Our result applies to some singular systems arising in the study of Hamiltonian systems with a strong restoring force and extends to a higher codimension the result obtained in our previous paper [4].

We study the existence of multiple positive solutions for a convex-concave problem, denoted by (P λμ ), with a nonlinear boundary condition involving two critical exponents and two positive parameters λ and μ. We obtain a continuous strictly decreasing function f such that K 1 ≡ {(f(μ), μ) : μ ∈ [0, ∞)} divides [0, ∞) × [0, ∞) \ {(0, 0)} in two connected sets K 0 and K 2 such that problem (P λμ ) has at least two solutions for (λ, μ) ∈ K 2 , at least one solution for (λ, μ) ∈ K 1 and no solution for (λ, μ) ∈ K 0 .