Entire solutions in ${\mathbb{R}}^{2}$ for a class of Allen-Cahn equations

Abstract

We consider a class of semilinear elliptic equations of the form −ε 2 ∆u(x, y) + a(x)W (u(x, y)) = 0, (x, y) ∈ R 2 (0.1) where ε > 0, a : R → R is a periodic, positive function and W : R → R is modeled on the classical two well Ginzburg-Landau potential W (s) = (s 2 − 1) 2. We look for solutions to (0.1) which verify the asymptotic conditions u(x, y) → ±1 as x → ±∞ uniformly with respect to y ∈ R. We show via variational methods that if ε is sufficiently small and a is not constant, then (0.1) admits infinitely many of such solutions, distinct up to translations, which do not exhibit one dimensional symmetries.

Brake orbits type solutions to some class of semilinear elliptic equations

“…First, thanks to the discreteness of the set {F ≤ c * } described by (2.15), as in [6] (see Lemma 3.1) we obtain that if ϕ p (u) < +∞ then, the trajectory y ∈ R → u(·, y) ∈ is bounded. Precisely Lemma 3.2 There exists C > 0 such that if u ∈ M p and (y 1 , y 2 ) ⊂ R then u(·, y 1 ) − u(·, y 2 ) ≤ C ϕ p (u).…”

“…its Lagrangian does not explicitly depend on y. This implies, as shown in [6], that if u ∈ H is a solution to (1.2) then the Energy function…”

“…Pursuing the study already faced in [4,6,7] we consider in this paper the problem of multiplicity of solutions to the boundary value problem…”

“…To prove Theorem 1.1 we make use of variational methods and we apply an Energy constrained variational Principle already introduced and used in [6,7]. Given c p ∈ (c, c * ), regular for F, we look for minima of the renormalized functional…”

Saddle solutions for bistable symmetric semilinear elliptic equations

“…To prove Theorem 1.1 we develop a global variational procedure, inspired to the ones introduced in [3,7], which allows us to recover the saddle type solution θ 3 from the minimum of a suitably renormalized action functional (for the use of renormalized functionals in different contexts we also refer to [4][5][6]36] and to the comprehensive recent monograph [37]). We look for minima of the double renormalized functional…”