Abstract
In this paper, we study the concentration behavior of critical points with a minimax characterization to the following functional
constrain on \({S_c=\{u\in H^1({\mathbb{R}}^N)|~|u|_2=c,c > 0\}}\) when \({c\rightarrow (c^*)^+}\), where \({c^*=\left(2^{-1}b|Q|_2^{\frac{8}{N}} \right)^{\frac{N}{8-2N}}}\), \({N=1,2,3,}\) and \({Q}\) is up to translations, the unique positive solution of \({-2\Delta Q+\left(\frac{4}{N}-1\right)Q=|Q|^{\frac{8}{N}} Q}\) in \({{\mathbb{R}}^N}\).
As such constraint problem is \({L^2}\)-critical, it seems impossible to benefit from natural constraints \({V_c= \left\{u\in S_c|~a\int\limits_{{\mathbb{R}}^N}|\nabla u|^2+b\left(\ \int\limits_{{\mathbb{R}}^N}|\nabla u|^2\right)^2=\frac{2N}{N+4} \int\limits_{{\mathbb{R}}^N}|u|^{\frac{2N+8}{N}} \right\}}\). We show that the mountain pass energy level \({\gamma(c)=\inf\limits_{u\in M_c}I(u)}\) for some submanifold \({M_c\subset V_c}\) and then prove the strict monotonicity of \({\gamma(c)}\) on \({(c^*,+\infty)}\). We obtained that the critical point \({u_c}\) behaves like
for some \({y_c\in{\mathbb{R}}^N}\) as \({c}\) approaches \({c^*}\) from above.
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Partially supported by NSFC NO: 11501428, NSFC NO: 11371159.
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Ye, H. The mass concentration phenomenon for L 2-critical constrained problems related to Kirchhoff equations. Z. Angew. Math. Phys. 67, 29 (2016). https://doi.org/10.1007/s00033-016-0624-4
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DOI: https://doi.org/10.1007/s00033-016-0624-4