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The mass concentration phenomenon for L 2-critical constrained problems related to Kirchhoff equations

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Abstract

In this paper, we study the concentration behavior of critical points with a minimax characterization to the following functional

$$I(u)=\frac{a}{2} \int\limits_{{\mathbb{R}}^N}|\nabla u|^2+\frac{b}{4} \left(\ \int\limits_{{\mathbb{R}}^N}|\nabla u|^2\right)^2-\frac{N}{2N+8} \int\limits_{{\mathbb{R}}^N}|u|^{\frac{2N+8}{N}}$$

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

$$u_c(x)\approx\left(\frac{a^2}{2b(c^*)^2[(\frac{c}{c^*})^{\frac{8-2N}{N}}-1]^2} \right)^{\frac{N}{8}}Q\left(\left(\frac{a}{b(c^*)^2[(\frac{c}{c^*})^{\frac{8-2N}{N}}-1]} \right)^{\frac{1}{2}}(x-y_c)\right)$$

for some \({y_c\in{\mathbb{R}}^N}\) as \({c}\) approaches \({c^*}\) from above.

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Correspondence to Hongyu Ye.

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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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