Volume 7, Issue 4

We show that the problem has multiple solutions including sign-changing ones, where μ > 0, N ≥ 3, 1 < p < (N + 2)/(N − 2) and Q : ℝ N → ℝ is some continuous function. We obtain signchanging solutions which are not only local minimum types in N * but also mountain pass types in N * , where N * = {u = u + +u − ∈ H 1 (ℝ N ) : u + ≠ 0, u − ≠ 0, ∫ ℝN (|∇u ± | 2 + μ|u ± | 2 ) dx = ∫ ℝN Q(x)|u ± | p+1 dx}.

Periodic bounce solutions of the second order Hamiltonian system −x″ = ∇V (x, t), x(t) ∈ Ω̅ is studied, where Ω ⊂ ℝ N is a bounded domain diffeomorphic to the unit ball and V ∈ C 2 (Ω̅ × ℝ) is periodic in the time t.

Following [3] we study the following minimization problem: in any dimension n ≥ 4 and under suitable assumptions on a(x). Mainly we assume that a(x) belongs to the Lorentz space L n/2 ,d(Ω) and the set N ≡ {x ∈ Ω|a(x) < 0} has positive Lebesgue measure. Notice that this last condition is satisfied when the set N has a nontrivial interior part (in fact this is the typical assumption imposed in the literature on the set N).

In this paper, we prove some existence theorems for hypersurfaces of prescribed k-order mean curvature in exterior domains, which lead to the study of the existence of k-order mean curvature equation in exterior domains.

This paper is concerned with the existence of positive solutions for a class of quasilinear elliptic equations on ℝ. The results are proved by combining the concentration-compactness principle due to Lions with a minimization approach.

We look for positive solutions to the nonlinear Schrödinger equation −Ɛ 2 ∆u − V (x)f′(u) = 0 in ℝ N , where V is a continuous bounded positive potential and f satisfies particular growth conditions which make our problem fall in the so called “zero mass case”. We prove an existence result for any Ɛ > 0, and a multiplicity result for Ɛ sufficiently small.

We study the singular periodic boundary value problem of the form (|u′| p−2 u′)′ = f(t, u), u(0) = u(T), u′(0) = u′(T), where 1 < p < ∞ and f ∈ Car([0, T] × (0,∞)) can have a repulsive space singularity at x = 0. Contrary to previous results by Mawhin and Jebelean, Liu Bing and Rachůnková and Tvrdý, we need not assume any strong force conditions. Our main existence results rely on a new antimaximum principle for periodic quasilinear periodic problem, which has an independent meaning.

We establish the global weak solutions to quantum drift-diffusion model, a fourth order parabolic system, in two or there space dimensions with large initial value and periodic boundary conditions and furthermore obtain the semiclassical limit by entropy estimate and compactness argument.

We discuss existence and multiplicity of solutions of the one-dimensional autonomous prescribed curvature problem , u(0) = 0, u(1) = 0, depending on the behaviour at the origin and at infinity of the function f. We consider solutions that are possibly discontinuous at the points where they attain the value zero.