Normalized bound states for the nonlinear Schrödinger equation in bounded domains

Normalized bound states for the nonlinear Schrödinger equation in bounded domains Given $$\rho >0$$ ρ > 0 , we study the elliptic problem $$\begin{aligned} \text {find } (U,\lambda )\in H^1_0(\Omega )\times {\mathbb {R}}\text { such that } {\left\{ \begin{array}{ll} -\Delta U+\lambda U=|U|^{p-1}U\\ \int _{\Omega } U^2\, dx=\rho , \end{array}\right. } \end{aligned}$$ find ( U , λ ) ∈ H 0 1 ( Ω ) × R such that - Δ U + λ U = | U | p - 1 U ∫ Ω U 2 d x = ρ , where $$\Omega \subset {\mathbb {R}}^N$$ Ω ⊂ R N is a bounded domain and $$p>1$$ p > 1 is Sobolev-subcritical, searching for conditions (about $$\rho $$ 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, i.e. $$1<p<1+4/N$$ 1 < p < 1 + 4 / N , the problem admits solutions for every $$\rho >0$$ ρ > 0 . In the $$L^2$$ L 2 -critical and supercritical case, i.e. when $$1+4/N \le p < 2^*-1$$ 1 + 4 / N ≤ p < 2 ∗ - 1 , we show that, for any $$k\in {\mathbb {N}}$$ k ∈ N , the problem admits solutions having Morse index bounded above by k only if $$\rho $$ ρ is sufficiently small. Next we provide existence results for certain ranges of $$\rho $$ ρ , which can be estimated in terms of the Dirichlet eigenvalues of $$-\Delta $$ - Δ in $$H^1_0(\Omega )$$ 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Calculus of Variations and Partial Differential Equations Springer Journals

Normalized bound states for the nonlinear Schrödinger equation in bounded domains

Loading next page...
 
/lp/springer_journal/normalized-bound-states-for-the-nonlinear-schr-dinger-equation-in-8zp9ngCUFJ
Publisher
Springer Berlin Heidelberg
Copyright
Copyright © 2017 by Springer-Verlag GmbH Germany
Subject
Mathematics; Analysis; Systems Theory, Control; Calculus of Variations and Optimal Control; Optimization; Theoretical, Mathematical and Computational Physics
ISSN
0944-2669
eISSN
1432-0835
D.O.I.
10.1007/s00526-017-1232-7
Publisher site
See Article on Publisher Site

References

You’re reading a free preview. Subscribe to read the entire article.


DeepDyve is your
personal research library

It’s your single place to instantly
discover and read the research
that matters to you.

Enjoy affordable access to
over 12 million articles from more than
10,000 peer-reviewed journals.

All for just $49/month

Explore the DeepDyve Library

Unlimited reading

Read as many articles as you need. Full articles with original layout, charts and figures. Read online, from anywhere.

Stay up to date

Keep up with your field with Personalized Recommendations and Follow Journals to get automatic updates.

Organize your research

It’s easy to organize your research with our built-in tools.

Your journals are on DeepDyve

Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more.

All the latest content is available, no embargo periods.

See the journals in your area

Monthly Plan

  • Read unlimited articles
  • Personalized recommendations
  • No expiration
  • Print 20 pages per month
  • 20% off on PDF purchases
  • Organize your research
  • Get updates on your journals and topic searches

$49/month

Start Free Trial

14-day Free Trial

Best Deal — 39% off

Annual Plan

  • All the features of the Professional Plan, but for 39% off!
  • Billed annually
  • No expiration
  • For the normal price of 10 articles elsewhere, you get one full year of unlimited access to articles.

$588

$360/year

billed annually
Start Free Trial

14-day Free Trial