Existence of a nontrivial solution for a strongly indefinite periodic Choquard system
Existence of a nontrivial solution for a strongly indefinite periodic Choquard system
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DOI：
10.1007/s00526-014-0797-7
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发表时间：
2015-09
影响因子：
2.1
通讯作者：
Shaowei Chen;L. Xiao
中科院分区：
文献类型：
--
作者：
Shaowei Chen;L. Xiao
考虑Choquard系统{-Δ u+ V（x）u+| u| ^ p-2 u= λ u，&\quad in R^ 3，\-Δ = u^ 2，&\quad in R^ 3. -Δ u+ V（x）u+| u| p-2 u= λ <$u，在R 3中，-Δ <$= u 2，在R 3中。其中λ> 0 λ> 0是一个参数，3< p< 6 3< p< 6，V ∈ C（R^3）V∈ C（R3）在x_jxj中是11-周期的，j= 1，2，3 j= 1，2，3，且0在算子-Δ+ V-Δ+ V的谱隙中。算子-Δ+ V-Δ+ V具有无穷维的负空间和正空间，并且具有非线性的竞争相互作用|u| ^ p-2 u| u| p-2 u和λ <$u λ <$u。此外，该系统对应的泛函不满足Palais-Smale条件。利用一个新的无穷维连接定理，证明了当λ> 0，λ> 0充分小时，该系统存在非平凡解.
We consider the Choquard system {-Δ u+ V (x) u+| u|^ p-2 u= λ ϕ u, &\quad in R^ 3,\-Δ ϕ= u^ 2, &\quad in R^ 3..-Δ u+ V (x) u+| u| p-2 u= λ ϕ u, in R 3,-Δ ϕ= u 2, in R 3. where λ> 0 λ> 0 is a parameter, 3< p< 6 3< p< 6, V ∈ C (R^ 3) V∈ C (R 3) is 1 1-periodic in x_j xj for j= 1, 2, 3 j= 1, 2, 3 and 0 is in a spectral gap of the operator-Δ+ V-Δ+ V. This system is strongly indefinite, ie, the operator-Δ+ V-Δ+ V has infinite-dimensional negative and positive spaces and it has a competitive interplay of the nonlinearities| u|^ p-2 u| u| p-2 u and λ ϕ u λ ϕ u. Moreover, the functional corresponding to this system does not satisfy the Palais–Smale condition. Using a new infinite-dimensional linking theorem, we prove that, for sufficiently small λ> 0, λ> 0, this system has a nontrivial solution.