Volume 3, Issue 2

In this paper we are concerned with the existence of solutions for the Hamiltonian system where N ≥ 3, Q, K : ℝ N → ℝ are bounded positive continuous functions and the numbers p, q > 1 satisfy Using dual variational methods we show that under certain conditions on Q and K the system (S ∊ ) has a positive solution which concentrates as ∊ → 0 in a point x o ∈ ℝ N where related functionals realize its minimum energy.

We study the Sobolev trace embedding W 1,p (Ω) ↪ L q (∂Ω), looking at the dependence of the best constant and the extremals on p and q. We prove that there exists a uniform bound (independent of (p, q)) for the best constant if and only if (p, q) lies far from (N, ∞). Also we study some limit cases, q = ∞ with p > N or p = ∞ with 1 ≤ q ≤ ∞.

In this work we consider positive solutions to cooperative elliptic systems of the form -Δu = λu - u 2 + buυ, -Δυ = μυ - υ 2 + cuυ a bounded smooth domain Ω ⊂ ℝ N (λ, μ ∈ ℝ, b, c > 0) which blow up on the boundary ∂Ω, that is u(x), v(x) → ∞ as dist(x, ∂Ω) → 0. We show existence and nonexistence of solutions, and give sufficient conditions for uniqueness. We also provide an exact estimate of the behaviour of the solutions near the boundary in terms of dist(x, ∂Ω).

We consider a 1-parameter family of hyperbolic surfaces M(t) of genus ν which degenerate as t → 0 and we obtain a precise estimate of λ 1,p (t), the first eigenvalue of the p-Laplacian (p > 1) on M(t). In some cases we also give a precise estimate of the first eigenfunctions. As a direct application, we obtain that the quotient (which is invariant under scaling of the metric) is unbounded even on the set of Riemannian manifolds with constant sectional curvature. This is to our knowledge, the first example of a family of manifolds with this property. To prove our results we use in an essential way the geometry of hyperbolic surfaces which is very well known. We show that an eigenfunction for λ 1,p (t) of L p norm one is almost constant in the L p sense (as t → 0) on the parts of M(t) with large injectivity radius, and we estimate precisely its p-energy on the parts with small injectivity radius.

This work deals with the nonexistence of solutions of the semilinear elliptic equation -Δu = a(x)|u| p in ℝ N + = { x = ( x 1 , . . . , x N ) ∈ ℝ N : x N > 0 ) , with the boundary condition on ∂ℝ N + = {(xʹ, 0) : xʹ ∈ ℝ N-1 }, where a(x) ≥ 0, b(xʹ) ≥ 0, p > 1 and q > 1. We are also concerned with a nonexistence result for the system -Δu i = a i (x)|u i+1 | pi , x ∈ ℝ N + , 1 ≤ i ≤ m, u m+1 = u 1 with on ∂ℝ N + , where a i (x) ≥ 0 and p i > 1.

Our purpose is to find positive solutions u ∈ D 1,2 (ℝ N ) of the semilinear elliptic problem -Δu - λV(x)u = h(x)u p-1 for 2 < p. The functions V and h may have an indefinite sign and the linearized operator need not have a fist (principal) eigenvalue, e.g. we allow V ≡ 1. We give precise existence and nonexistence criteria, which depend on λ and on the growth of h - and h + /V + . Existence theorems are obtained by constrained minimization. The mountain pass theorem leads to a second solution.

In this paper we study a semilinear elliptic equation in all ℝ N . This equation depends on a parameter λ, and we obtain, for small λ < 0, solutins which are small in H 1 (ℝ N ). In this sense we have solutions bifurcating from the origin and, as the differential operator involved is the laplacian, we say that we have solutions bifurcating from the bottom of the essential spectrum of the laplacian. By a change of variables we transform the original bifurcation problem into a perturbation one. We adopt a variational procedure, looking for critical points of a suitable functional. We apply a recently developped reduction method, which allows to reduce the original variational problem in H 1 (ℝ N ) to a variational problem in a finite-dimensional manifold, and then we solve this last problem. In this way we are also able to manage the presence of critical nonlinearities, in the sense of Sobolev embedding.

We prove the uniqueness of positive W 0 1,2 solutions for the following problem, which involves the singular potential |x| -2 : where N ≥ 3, 1 < p < , and 0 < λ < . We also give a complete description of the set of positive singular solutions of the problem.

We revisit the question of exact multiplicity of positive solutions for a class of Dirichlet problems for cubic-like nonlinearities, which we studied in [6]. Instead of computing the direction of bifurcation as we did in [6], we use an indirect approach, and study the evolution of turning points. We give conditions under which the critical (turning) points continue on smooth curves, which allows us to reduce the problem to the easier case of f (0) = 0. We show that the smallest root of f (u) does not have to be restricted.