Normalized bound states for the nonlinear Schrödinger equation in bounded domains
Normalized bound states for the nonlinear Schrödinger equation in bounded domains
复制标题
DOI：
10.1007/s00526-017-1232-7
复制
发表时间：
2016-07
影响因子：
2.1
通讯作者：
D. Pierotti;G. Verzini
中科院分区：
文献类型：
--
作者：
D. Pierotti;G. Verzini
给定ρ> ρ> 0，研究椭圆型问题find (U, λ)∈H^ 1_0 （Ω） * R使得\left {find (U, λ)∈H^ 1_0 （Ω）× R使得-Δ U+ λ U=| U| p-1 U∫Ω U 2 dx= ρ，其中Ω∧R^ N Ω∧RN是有界域，p> 1 p> 1是sobolevv -次临界，寻找解存在的条件（关于ρ ρ， N和p）。由伽利亚多-尼伦伯格不等式可知，当p为L^ 2l2次临界时，即1< p< 1+ 4/N 1< p< 1+ 4/N，问题对每一个ρ> 0 ρ> 0都有解。在L^ 2 l2临界和超临界情况下，即当1+ 4/N≤p< 2^*-1 1+ 4/N≤p< 2 *-1，我们证明了，对于任何k∈N k∈N，只有当ρ ρ足够小时，问题才允许有莫尔斯指数以k为界的解。接下来，我们给出了ρ ρ在H^ 1_0 （Ω） H 0 1 （Ω）中的Dirichlet特征值-Δ-Δ在一定范围内的存在性结果，并将其推广到变符号解和在Anal中Noris等人得到的一些结果。PDE 7: 1807-1838, 2014。
Abstract Given ρ> 0 ρ> 0, we study the elliptic problem find (U, λ) ∈ H^ 1_0 (Ω) * R such that\left {find (U, λ)∈ H 0 1 (Ω)× R such that-Δ U+ λ U=| U| p-1 U∫ Ω U 2 dx= ρ, where Ω ⊂ R^ N Ω⊂ RN is a bounded domain and p> 1 p> 1 is Sobolev-subcritical, searching for conditions (about ρ ρ, N and p) for the existence of solutions. By the Gagliardo-Nirenberg inequality it follows that, when p is L^ 2 L 2-subcritical, ie 1< p< 1+ 4/N 1< p< 1+ 4/N, the problem admits solutions for every ρ> 0 ρ> 0. In the L^ 2 L 2-critical and supercritical case, ie when 1+ 4/N ≤ p< 2^*-1 1+ 4/N≤ p< 2∗-1, we show that, for any k ∈ N k∈ N, the problem admits solutions having Morse index bounded above by k only if ρ ρ is sufficiently small. Next we provide existence results for certain ranges of ρ ρ, which can be estimated in terms of the Dirichlet eigenvalues of-Δ-Δ in H^ 1_0 (Ω) H 0 1 (Ω), extending to changing sign solutions and to general domains some results obtained in Noris et al. in Anal. PDE 7: 1807–1838, 2014 for positive solutions in the ball.