Many Solutions of Elliptic Problems on Rⁿof Irrational Slope

Abstract

We consider the problem − u + F u x u = 0 on R n , where F is a smooth function periodic of period 1 in all its variables. We show that, under suitable hypotheses on F , this problem has a family of non-self-intersecting solutions u D , which are at finite distance from a plane of slope 0 0 with irrational. These solutions depend on a real parameter D; if D = D , then the closures of the integer translates of u D and of u D do not intersect.

Existence and Modulation of Uniform Sliding States in Driven and Overdamped Particle Chains

“…One conclusion obtained in [13] is that either there is a critical point with strictly positive renormalized energy or there is a foliation of Birkhoff minimizers, which is similar to our discussion on the relation of the depinning force and the ground states. Related results in the elliptic integrand setting have been obtained by Bessi [9] and Matano and Rabinowitz [23]. One method used in [9,13] is to consider the heat flow associated with the elliptic equation, just as we consider (1.5) to study the equilibrium states.…”

“…One method used in [9,13] is to consider the heat flow associated with the elliptic equation, just as we consider (1.5) to study the equilibrium states. So the related arguments in this paper might have some applications in the construction of oscillatory solutions for partial differential equations, see [9,13,23].…”

Bump solutions for the mesoscopic Allen–Cahn equation in periodic media

“…Of course, "| · |" is here the Lebesgue measure on T n . We let U (2) ε ⊂ T n be an open set such that U (1) ε ⊂ U (2) ε and |U (2) ε | = ε/2, and we take a partition of unity ψ ε ∈ C ∞ (T n , [0, 1]), with ψ ε ≡ 1 on U (1) ε and ψ ε ≡ 0 outside U (2) ε . Let also ρ ε be a convolution kernel such that…”

Single and Multitransition Solutions for a Family of Semilinear Elliptic PDE’s

“…Bessi [15]- [16] has constructed solutions of (PDE) that shadow a family of solutions possessing different rotation vectors and therefore do not possess this hyperplane property. See also [9] for another result of this type.…”