Piero Montecchiari

EXISTENCE OF INFINITELY MANY STATIONARY LAYERED SOLUTIONS IN R 2 FOR A CLASS OF PERIODIC ALLEN–CAHN EQUATIONS

ABSTRACT We consider a class of periodic Allen–Cahn equations where is an even, periodic, positive function and is modeled on the classical two well Ginzburg–Landau potential . We show, via variational methods, that there exist infinitely many solutions, distinct up to periodic translations, of 1 asymptotic as to the pure states ±b, i.e., solutions satisfying the boundary conditions In fact, we prove the existence of solutions of 1-2 which are periodic in the y variable and if such solutions are finite modulo periodic translations, we can prove the existence of infinitely many (modulo periodic translations) solutions of 1-2 asymptotic to different periodic solutions as .

Multiplicity of Entire Solutions For a Class of Almost Periodic Allen-Cahn Type Equations

Abstract We consider a class of semilinear elliptic equations of the form −Δu(x,y) + a(εx)Wʹ(u(x,y)) = 0, (x,y) ∊ ℝ2 (0.1) where ε > 0, a : ℝ → ℝ is an almost periodic, positive function and W : ℝ → ℝ is modeled on the classical two well Ginzburg-Landau potential W(s) = (s2 - 1)2. We show via variational methods that if ε is sufficiently small and a is not constant then (0.1) admits infinitely many two dimensional entire solutions verifying the asymptotic conditions u(x, y) → ±1 as x → ±∞ uniformly with respect to y ∊ ℝ.

On the existence of homoclinic orbits for the asymptotically periodic Duffing equation

where p > 1 and a : R → R satisfies: (a1) a ∈ L∞(R), inf a > 0, (a2) a = a∞ + a0, with a∞ T -periodic and a0(t)→ 0 as t→ ±∞. Noting that 0 is a hyperbolic rest point for (1.1), we look for homoclinic orbits to 0, namely non trivial solutions to (1.1) such that u(t) → 0 and u(t) → 0 as t→ ±∞. The homoclinic problem for equation (1.1), possibly with a more general nonlinearity, as well as the analogous subcritical elliptic problem on R, has been successfully studied with variational methods by several authors, for different kinds of behaviour of the coefficient a.

An energy constrained method for the existence of layered type solutions of NLS equations

Abstract We study the existence of positive solutions on R N + 1 to semilinear elliptic equation − Δ u + u = f ( u ) where N ⩾ 1 and f is modeled on the power case f ( u ) = | u | p − 1 u . Denoting with c the mountain pass level of V ( u ) = 1 2 ‖ u ‖ H 1 ( R N ) 2 − ∫ R N F ( u ) d x , u ∈ H 1 ( R N ) ( F ( s ) = ∫ 0 s f ( t ) d t ), we show, via a new energy constrained variational argument, that for any b ∈ [ 0 , c ) there exists a positive bounded solution v b ∈ C 2 ( R N + 1 ) such that E v b ( y ) = 1 2 ‖ ∂ y v b ( ⋅ , y ) ‖ L 2 ( R N ) 2 − V ( v b ( ⋅ , y ) ) = − b and v ( x , y ) → 0 as | x | → + ∞ uniformly with respect to y ∈ R . We also characterize the monotonicity, symmetry and periodicity properties of v b .