An elementary construction of complex patterns in nonlinear Schr$ouml$dinger equations

Abstract

We consider the problem of finding standing waves to a nonlinear Schrödinger equation. This leads to searching for solutions of the equationby isolated local maxima or minima of V , and corresponding arbitrary integers n i , i = 1, . . . , m, we prove that there is a finite energy solution exhibiting a cluster of n i spikes concentrating around each x i as ε → 0. The clusters consist of spikes with alternating sign if the point is a minimum, and of constant sign if it is a maximum. This construction extends to infinitely many clusters of spikes under appropriate conditions. The proof follows an elementary variational matching approach, which resembles the so-called broken-geodesic method.

The Toda System and Clustering Interfaces in the Allen–Cahn equation

“…In the one dimensional or radial cases, concentration in the form of clusters of transition layers or spikes has already been observed in various problems in the literature, see [4,10,11,13,32,33,27,28], phenomena in accordance with higher-dimensional multi-spike clustering as predicted in [20]. In particular, multiple radial transition spheres collapsing on the boundary have been found for Allen-Cahn in [28].…”

Sign-changing multi-bump solutions for nonlinear Schrödinger equations with steep potential wells

“…Isolatedness of critical values of Ψ 1,D (v 1 ) corresponding to positive solutions is important, and we will construct a deformation flow for J λ ∈ C 1 (Σ 1,λ × Σ 2,λ , R), under which the level set {v 1 ∈ Σ 1,λ ; J 1,λ (v 1 ) ≤ c} for the unperturbed functional J 1,λ (v 1 ) is invariant for a certain range of c. In Section 5 we will give a proof to Theorem 1.2 via Nehari type methods (cf. del Pino, Felmer and Tanaka [15]). In Section 6, we give a proof to our Theorems 1.3, 1.5.…”

“…We use the broken-geodesic method which is used in [15]. See also [19] and [26] for related arguments.…”

“…The following lemma deals with (P λ,i ) (i = 0, 2, 4), which can be shown just as in [15]. (t 1 , t 2 , t 3 , t 4 ) ∈ U , (i) (P λ,0 ) has a unique solution u 0,λ (t 1 ; x) which satisfies…”

Positive and sign-changing clusters around saddle points of the potential for nonlinear elliptic problems

“…After that, several papers have addressed the question of existence of multibump solutions concentrating around a minimum of V . This result has become known first for exactly one positive and one negative peak ( [1], [5]), and later under polygonal symmetries of V ( [8]), or in the one-dimensional case ( [13]). In a recent paper clusters with at most 6 mixed positive and negative peaks have been found, see [9].…”