MULTIBUMP SOLUTIONS AND CRITICAL GROUPS

作者：Arioli Gianni^*; Szulkin Andrzej; Zou Wenming

来源：Transactions of the American Mathematical Society, 2009, 361(6): 3159-3187.

DOI：10.1090/s0002-9947-09-04669-8

• Multibump solution
• critical group
• Bernoulli shift
• Newtonian system
• Schrodinger equation
• critical exponent

We consider the Newtonian system -q B(t)q = W(q)(q, t) with B, W periodic in t, B positive definite, and show that for each isolated homoclinic solution q(0) having a nontrivial critical group ( in the sense of Morse theory), multibump solutions ( with 2 <= k <= infinity bumps) can be constructed by gluing translates of q(0). Further we show that the collection of multibumps is semiconjugate to the Bernoulli shift. Next we consider the Schrodinger equation -Delta u V(x)u = g(x, u) in R(N), where V, g are periodic in x(1),..., x(N), sigma(-Delta V) subset of (0, infinity), and we show that similar results hold in this case as well. In particular, if g(x, u) = vertical bar u vertical bar(2)*(-2)u, N >= 4 and V changes sign, then there exists a solution minimizing the associated functional on the Nehari manifold. This solution gives rise to multibumps if it is isolated.

• 出版日期2009
• 单位清华大学