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The existence of normalized solutions for $L^2$-critical quasilinear Schrödinger equations
  • Hongyu Ye,
  • Yingying Yu
Hongyu Ye
Wuhan University of Science and Technology
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Yingying Yu
Wuhan University of Science and Technology
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Abstract

In this paper, we study the existence of critical points for the following functional $$I(u)=\frac{1}{2}\ds\int_{\R^N}|\nabla u|^2+\ds\int_{\R^N}|u|^2|\nabla u|^2-\frac{N}{4(N+1)}\ds\int_{\R^N}|u|^{\frac{4(N+1)}{N}},$$ constrained on $S_c=\{u\in H^1(\R^N)|~\int_{\R^N}|u|^2|\nabla u|^2<+\infty,~|u|_2=c,c>0\}$, where $N\geq1$. The constraint problem is $L^2$-critical. We prove that the minimization problem $i_c=\inf\limits_{u\in S_c}I(u)$ has no minimizer for all $c>0$. We also obtain a threshold value of $c$ separating the existence and nonexistence of critical points for $I(u)$ restricted to $S_c$.