Saddle-shaped positive solutions for elliptic systems with bistable nonlinearity

Abstract

In this paper we prove the existence of infinitely many saddle-shaped positive solutions for non-cooperative nonlinear elliptic systems with bistable nonlinearities in the phaseseparation regime. As an example, we prove that the systemhas infinitely many saddle-shape solutions in dimension 2 or higher. This is in sharp contrast with the case Λ ∈ (0, 1], for which, on the contrary, only constant solutions exist.

Minimizing under relaxed symmetry constraints: Triple and $N$-junctions

“…From (1.14) it follows that, if h = 2, the solution U in Theorem 1.3 is saddle shaped. Existence of saddle solution of (1.13) in R 2 equivariant with respect to Z 4 , the reflection group of the symmetries of the square was first established in [8] and generalized to the case of equivariance with respect to Z 2N in [1], see also [17]. Existence and stability of saddle shaped solutions of (1.13) in R 2n was discussed in [7].…”