The existence and asymptotic behavior of normalized solutions for Kirchhoff equation with singular potential

In this paper, the existence, non-existence and asymptotic behavior of minimizers for a constrained Kirchhoff type minimization problem with singular potential were considered. Using concentration-compactness lemma, the existence of minimizers of this minimization problem was obtained. By truncation function technique, the non-existence of minimizers was got. Compared with the known result for this problem with trapping potential (Guo, Zhang, Zhou, 2018, [4]) that there exists minimizer for the case of critical exponent p=p⁎ and critical parameter β=βp⁎, our problem has no minimizers in the same condition. Therefore, we discuss the blow-up behavior of minimizer for p=p⁎ as β approaches βp⁎ or β=β˜p and p approaches p⁎, and they appear to be new.