Liouville-type theorems for elliptic Schrödinger systems associated with copositive matrices

Philippe Souplet; Philippe Souplet

doi:10.3934/nhm.2012.7.967

Networks and Heterogeneous Media

2012, Volume 7, Issue 4: 967-988. doi: 10.3934/nhm.2012.7.967

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Liouville-type theorems for elliptic Schrödinger systems associated with copositive matrices

Philippe Souplet ^{1
,}

1.
Université Paris 13, Sorbonne Paris Cité, Laboratoire Analyse, Géométrie et Applications, CNRS, UMR 7539, 93430 Villetaneuse

Received: 01 September 2011
Primary: 35J60, 35J47, 35B53; Secondary: 15B48.

We study non-cooperative, multi-component elliptic Schrödinger systems arising in nonlinear optics and Bose-Einstein condensation phenomena. We here reconsider the more delicate case of systems of $m ≥ 3$ components. We prove a Liouville-type nonexistence theorem in space dimensions $n ≥3$ for homogeneous nonlinearities of degree $ p < n/(n-2)$, under the optimal assumption that the associated matrix is strictly copositive. This extends recent work of Tavares, Terracini, Verzini and Weth [22] and of Quittner and the author [17], where the results were limited to $n ≤ 2$ dimensions or to $m=2$ components. The proof of the Liouville theorem is done by combining and improving different arguments from [22] and [17], namely a feedback procedure based on Rellich-Pohozaev type identities and functional analytic inequalities on $S^{n-1}$, and suitable test-function arguments. We also consider a more general class of systems with gradient structure, for which our arguments show the triviality of solutions satisfying a suitable integral bound, a result which may be of independent interest.
- Nonlinear elliptic system,
- Schrödinger system,
- nonexistence,
- Liouville-type theorem,
- strictly copositive matrices
Citation: Philippe Souplet. Liouville-type theorems for elliptic Schrödinger systems associated with copositive matrices[J]. Networks and Heterogeneous Media, 2012, 7(4): 967-988. doi: 10.3934/nhm.2012.7.967

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Abstract

We study non-cooperative, multi-component elliptic Schrödinger systems arising in nonlinear optics and Bose-Einstein condensation phenomena. We here reconsider the more delicate case of systems of $m ≥ 3$ components. We prove a Liouville-type nonexistence theorem in space dimensions $n ≥3$ for homogeneous nonlinearities of degree $ p < n/(n-2)$, under the optimal assumption that the associated matrix is strictly copositive. This extends recent work of Tavares, Terracini, Verzini and Weth [22] and of Quittner and the author [17], where the results were limited to $n ≤ 2$ dimensions or to $m=2$ components. The proof of the Liouville theorem is done by combining and improving different arguments from [22] and [17], namely a feedback procedure based on Rellich-Pohozaev type identities and functional analytic inequalities on $S^{n-1}$, and suitable test-function arguments. We also consider a more general class of systems with gradient structure, for which our arguments show the triviality of solutions satisfying a suitable integral bound, a result which may be of independent interest.

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