Brake orbits type solutions to some class of semilinear elliptic equations

Abstract

We consider a class of semilinear elliptic equations of the formwhere 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 show, via variational methods, that if the set of solutions to the one dimensional heteroclinic problemhas a discrete structure, then (0.1) has infinitely many solutions periodic in the variable y and verifying the asymptotic conditions u(x, y) → ±1 as x → ±∞ uniformly with respect to y ∈ R.

Stationary layered solutions for a system of Allen-Cahn type equations

“…Note that the Theorem guarantees the existence of a brake orbit type solution at level c whenever c ∈ (m, m + λ) is a regular value of V . As a consequence of the Sard Smale Theorem and the local compactness properties of V , it can be proved that the set of regular values of V is open and dense in [m, m + λ] (see Lemma 2.9 in [6]). Then, Theorem 1.2 provides in fact the existence of an uncountable set of geometrically distinct two dimensional solutions of (1.1) of brake orbit type.…”

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…”

Multiplicity of layered solutions for Allen–Cahn systems with symmetric double well potential

“…We finally refer to [2] where, adapting to the vectorial case an Energy constrained variational argument used in [5], [6], [7] [10], it is shown that (1.1)-(1.2) admits infinitely many planar solutions whenever the set of one dimensional minimal heteroclinic solutions is not connected. These planar solutions exhibit different behaviour with respect to the variable y, being periodic in y or asymptotic as y → ±∞ to one dimensional heteroclinic (not necessarily minimal) connections.…”