Abstract
We study the following Brézis–Nirenberg problem (Comm Pure Appl Math 36:437–477, 1983):
where Ω is a bounded smooth domain of R N (N ≧ 7) and 2* is the critical Sobolev exponent. We show that, for each fixed λ > 0, this problem has infinitely many sign-changing solutions. In particular, if λ ≧ λ1, the Brézis–Nirenberg problem has and only has infinitely many sign-changing solutions except zero. The main tool is the estimates of Morse indices of nodal solutions.
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Acknowledgments
W. Zou thanks Professor M. Ramos so much for many enlightening discussions when he was visiting CMAF (Lisboa, Portugal). Ramos’ suggestions are very highly appreciated. Schechter is supported by NSF. Zou is supported by NSFC (10871109, 10571096) and the program of the Ministry of Education in China for NCET in Universities of China.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Schechter, M., Zou, W. On the Brézis–Nirenberg Problem. Arch Rational Mech Anal 197, 337–356 (2010). https://doi.org/10.1007/s00205-009-0288-8
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DOI: https://doi.org/10.1007/s00205-009-0288-8