Abstract
In this paper, we study the existence of critical points for the following functional constrained on \({S_c=\{u\in H^1(\mathbb{R}^N)| |u|_2=c\}}\):
where N = 1, 2, 3 and a, b > 0 are constants. The constraint problem is L 2-critical. We showed that I(u) has a constraint critical point with a mountain pass geometry on S c if \({c > c^*:=(2^{-1}b|Q|_2^{\frac{8}{N}})^{\frac{N}{8-2N}}}\), where Q is the unique positive radial solution of \({-2\Delta Q+(\frac{4}{N}-1)Q=|Q|^{\frac{8}{N}} Q}\) in \({\mathbb{R}^N}\). For 0 < c < c *, I(u) has no critical point on S c , and we proved the existence of minimizers for a new perturbation functional on S c :
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Partially supported by NSFC No.: 11371159, NSFC No.: 11301204.
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Ye, H. The existence of normalized solutions for L 2-critical constrained problems related to Kirchhoff equations. Z. Angew. Math. Phys. 66, 1483–1497 (2015). https://doi.org/10.1007/s00033-014-0474-x
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DOI: https://doi.org/10.1007/s00033-014-0474-x