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The compactness of minimizing sequences for a nonlinear Schrödinger system with potentials

Department of Mathematics, Faculty of Science and Technology, Keio University, Yagami Campus, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan

E-mail Address: ikoma@math.keio.ac.jp

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Yasuhito Miyamoto

Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan

E-mail Address: miyamoto@ms.u-tokyo.ac.jp

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https://doi.org/10.1142/S0219199721501030Cited by:11 (Source: Crossref)

Abstract

In this paper, we consider the following minimizing problem with two constraints:

inf {E (u) | u = (u 1, u 2), ∥ u 1 ∥ 2 L 2 = α 1, ∥ u 2 ∥ 2 L 2 = α 2}, <math display="block" altimg="eq-00001.gif"><mrow><mo class="qopname">inf</mo><mfenced separators="" open="{" close="}"><mrow><mi>E</mi><mo class="MathClass-open" stretchy="false">(</mo><mi>u</mi><mo class="MathClass-close" stretchy="false">)</mo><mspace width=".17em" class="thinspace"></mspace><mo class="MathClass-rel">|</mo><mspace width=".17em" class="thinspace"></mspace><mi>u</mi><mo class="MathClass-rel">=</mo><mo class="MathClass-open" stretchy="false">(</mo><msub><mrow><mi>u</mi></mrow><mrow><mn>1</mn></mrow></msub><mo class="MathClass-punc">,</mo><msub><mrow><mi>u</mi></mrow><mrow><mn>2</mn></mrow></msub><mo class="MathClass-close" stretchy="false">)</mo><mo class="MathClass-punc">,</mo><mo class="MathClass-rel">∥</mo><msub><mrow><mi>u</mi></mrow><mrow><mn>1</mn></mrow></msub><msubsup><mrow><mo class="MathClass-rel">∥</mo></mrow><mrow><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><mn>2</mn></mrow></msubsup><mo class="MathClass-rel">=</mo><msub><mrow><mi>α</mi></mrow><mrow><mn>1</mn></mrow></msub><mo class="MathClass-punc">,</mo><mo class="MathClass-rel">∥</mo><msub><mrow><mi>u</mi></mrow><mrow><mn>2</mn></mrow></msub><msubsup><mrow><mo class="MathClass-rel">∥</mo></mrow><mrow><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><mn>2</mn></mrow></msubsup><mo class="MathClass-rel">=</mo><msub><mrow><mi>α</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></mfenced><mo class="MathClass-punc">,</mo></mrow></math>

where

$α_{1}, α_{2} > 0$ and

$E (u)$ is defined by

E(u):=∫RN{122∑i=1(|∇ui|2+Vi(x)|ui|2)−2∑i=1μi2pi+2|ui|2pi+2<math display="block" altimg="eq-00004.gif"><mrow><mi>E</mi><mo class="MathClass-open" stretchy="false">(</mo><mi>u</mi><mo class="MathClass-close" stretchy="false">)</mo><mo class="MathClass-punc">:</mo><mo class="MathClass-rel">=</mo><msub><mrow><mo class="MathClass-op">∫</mo></mrow><mrow><msup><mrow><mstyle mathvariant="bold"><mi>R</mi></mstyle></mrow><mrow><mi>N</mi></mrow></msup></mrow></msub><mfenced separators="" open="{" close=""><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><munderover accentunder="true" accent="true"><mrow><mo class="MathClass-op">∑</mo></mrow><mrow><mi>i</mi><mo class="MathClass-rel">=</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></munderover><mo class="MathClass-open" stretchy="false">(</mo><mo class="MathClass-rel">|</mo><mo class="MathClass-op">∇</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>i</mi></mrow></msub><msup><mrow><mo class="MathClass-rel">|</mo></mrow><mrow><mn>2</mn></mrow></msup><mo stretchy="false" class="MathClass-bin">+</mo><msub><mrow><mi>V</mi></mrow><mrow><mi>i</mi></mrow></msub><mo class="MathClass-open" stretchy="false">(</mo><mi>x</mi><mo class="MathClass-close" stretchy="false">)</mo><mo class="MathClass-rel">|</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>i</mi></mrow></msub><msup><mrow><mo class="MathClass-rel">|</mo></mrow><mrow><mn>2</mn></mrow></msup><mo class="MathClass-close" stretchy="false">)</mo><mo stretchy="false" class="MathClass-bin">−</mo><munderover accentunder="true" accent="true"><mrow><mo class="MathClass-op">∑</mo></mrow><mrow><mi>i</mi><mo class="MathClass-rel">=</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></munderover><mfrac><mrow><msub><mrow><mi>μ</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><mn>2</mn><msub><mrow><mi>p</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false" class="MathClass-bin">+</mo><mn>2</mn></mrow></mfrac><mo class="MathClass-rel">|</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>i</mi></mrow></msub><msup><mrow><mo class="MathClass-rel">|</mo></mrow><mrow><mn>2</mn><msub><mrow><mi>p</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false" class="MathClass-bin">+</mo><mn>2</mn></mrow></msup></mrow></mfenced></mrow></math>

−βp3+1|u1|p3+1|u2|p3+1}dx.<math display="block" altimg="eq-00005.gif"><mrow><mfenced separators="" open="" close="}"><mrow><mo stretchy="false" class="MathClass-bin">−</mo><mspace width=".17em" class="thinspace"></mspace><mfrac><mrow><mi>β</mi></mrow><mrow><msub><mrow><mi>p</mi></mrow><mrow><mn>3</mn></mrow></msub><mo stretchy="false" class="MathClass-bin">+</mo><mn>1</mn></mrow></mfrac><mo class="MathClass-rel">|</mo><msub><mrow><mi>u</mi></mrow><mrow><mn>1</mn></mrow></msub><msup><mrow><mo class="MathClass-rel">|</mo></mrow><mrow><msub><mrow><mi>p</mi></mrow><mrow><mn>3</mn></mrow></msub><mo stretchy="false" class="MathClass-bin">+</mo><mn>1</mn></mrow></msup><mo class="MathClass-rel">|</mo><msub><mrow><mi>u</mi></mrow><mrow><mn>2</mn></mrow></msub><msup><mrow><mo class="MathClass-rel">|</mo></mrow><mrow><msub><mrow><mi>p</mi></mrow><mrow><mn>3</mn></mrow></msub><mo stretchy="false" class="MathClass-bin">+</mo><mn>1</mn></mrow></msup></mrow></mfenced><mstyle mathvariant="italic"><mi>d</mi></mstyle><mi>x</mi><mo class="MathClass-punc">.</mo></mrow></math>

Here

$N \geq 1$ ,

$μ_{1}, μ_{2}, β > 0$ and

$V_{i} (x)$

$(i = 1, 2)$ are given functions. For

$V_{i} (x)$ , we consider two cases: (i) both of

$V_{1}$ and

$V_{2}$ are bounded, (ii) one of

$V_{1}$ and

$V_{2}$ is bounded. Under some assumptions on

$V_{i}$ and

$p_{j}$ , we discuss the compactness of any minimizing sequence.

Keywords:

AMSC: 35J50, 35J20, 35J61, 35Q55

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