The effect of the graph topology on the existence ofmultipeak solutions for nonlinear Schrödinger equations

E N Dancer,Shusen Yan,Kee Y Lam

doi:10.1155/s1085337501000276

E N Dancer, Shusen Yan + Show 1 more

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https://doi.org/10.1155/s1085337501000276

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Abstract

where V (y) is a smooth bounded function with positive lower bound, e > 0 is a small number, 2 2 and 2 < p <+∞ if N = 2. Many works have been done on problem (1.1) recently (cf. [6, 7, 8, 16, 21, 22, 23]). One of the results in the papers just mentioned is that if x1,x2, . . . ,xk are k different strictly local minimum points of V (y), then (1.1) has a k-peak solution ue, that is, solution with exactly k local maximum points, such that ue has exactly one local maximum point in a neighborhood of xj , j = 1, . . . ,k. The same conclusion is also true if x1,x2, . . . ,xk are k different strictly local maximum points of V (y). Actually, it is proved in [23] that (1.1) has a multipeak solution with all its peaks near an isolated maximum point of V (y). Thus a natural question is what will happen if V (y) attains its local minimum or local maximum on a connected set. Especially, if V (y) attains its local minimum on a connected set which contains infinitely many points, it is interesting to study whether (1.1) has multipeak solution concentrating on this set. Generally, this is not true as shown in Example 1.6. The main results of this paper consist of three parts. First, we study how the topological structure of the local minimum set of the potential V (y) affects the existence of multipeak solutions for (1.1). We show that if the minimum set of V (y) has nontrivial reduced homology, then for each k ≥ 1, (1.1) has at least

Highlights

Consider the problem−ε2 u + V (y)u = up−1, y ∈ RN, u > 0, y ∈ RN, (1.1)u −→ 0, as |y| −→ +∞, where V (y) is a smooth bounded function with positive lower bound, ε > 0 is a small number, 2 < p < 2N/(N − 2) if N > 2 and 2 < p < +∞ if N = 2.Many works have been done on problem (1.1) recently
A natural question is what will happen if V (y) attains its local minimum or local maximum on a connected set
If V (y) attains its local minimum on a connected set which contains infinitely many points, it is interesting to study whether (1.1) has multipeak solution concentrating on this set

Summary

Introduction

Let M be a connected compact local minimum or maximum set. (1.1) has at least N different k-peak solutions concentrating on the connected compact local maximum set of V (y). (i) equation (1.16) has at least N boundary k-peak solutions with all their local maximum points near the global minimum set of the mean curvature function of ∂ ;. We apply Proposition 2.3 to prove that for ε > 0 small, K(x) has a critical point in Kcε,2 \ Kcε,. Where Tτ = ∪i=j {|xi − xj | ≤ τ, xi, xj ∈ Mεα }, C > c > 0 are some suitable constants It follows from Lemma A.2 that cε,1 = εN kVMp/(p−2)−N/2A − T εN+αh > K(x).

Multipeak solutions concentrating on the maximum set

Multipeak solution concentrating on different sets

Some topological results

Some estimates of the cuplength