Volume 2, Issue 1

The aim of this paper is to study the existence of a nontrivial solution of the following semilinear elliptic variational inequality where Ω is an open bounded subset of ℝ N (N ≥ 1), λ is a real parameter, with λ ≥ λ 1 ,the first eigenvalue of the operator - Δ in H 0 1 (Ω), ψ belongs to H 1 (Ω), ψ |∂Ω ≥ 0and p is a Carathéodory function on Ω ×ℝ, which satisfies some general superlinearity growth conditions at zero and at infinity.

Establishing some general results on the principal eigenvalues for a class of multipoint boundary value problems, we prove existence and uniqueness theorems for positive solutions of nonlinear multipoint boundary value problems. Our argument is based mainly on the maximum principle [4] (highlighting new ordered Banach spaces of continuous functions) and on the continuity of the spectral radius mapping for compact linear operators.

We discuss the critical exponent problem for the semilinear wave equation with linear damping u tt +(-1) m |x| α Δ m u + u t = f(t,x)|u| p + w(t,x), t > 0, x ∈ℝ n ,and show that it coincides with the Fujita exponent for the heat equation with the same nonlinearity.

A new result about the existence of a closed geodesic on a Riemannian manifold with boundary is given. A detailed comparison with previous results is carried out.

In this paper we study a class of field equations, in several space dimensions, which admits solitary waves. The equation is a vector-valued version of a field equation proposed by Derrick in 1964 as model for elementary particles. We show the existence of infinitely many solutions with arbitrary topological charge.

In this paper we discuss the existence and nonexistence of minimizer for the constraint minimization problem where r = |x′| for x = (x′, z) ∊ ℝ k × ℝ n-k , 2 ≤ k ≤ n, 1 < p < n, 0 ≤ s < p, s < k and . We obtain conditions under which the above problem admits a minimizer.