부산대학교 도서관

This paper concerns the following Kirchhoff equation:$$ \cases - \left( a + b \int_{\Omega} |\nabla u|^2 \,dx \right) \Delta u = \muu + \beta |u|^{p} u + \lambda |u|^q u, &\quad x\in \Omega,\\u=0, &\quad x \in \partial \Omega,\endcases$$where $\Omega\subset\Bbb R^3$ is a bounded domain, $p \in [\frac{8}{3},4)$, $q \in (0,\frac{8}{3})$, $a$, $b>0$ are constants, ${\beta,\lambda >0}$ denote parameters and $\mu$ is a Lagrange multiplier. Weare interested in the following minimization problem:$$e(\beta, \lambda) \coloneq \inf_{u \in S_1} E(u),\tag1$$where $E$ is the energy functional associated with (1) and $S_1= \{ u\in H^1_0(\Omega) \mid \int_{\Omega} |u|^2 \,dx =1 \}$. The existenceand nonexistence of constraint minimizers of (1) are closely relatedwith the following elliptic problem:$$- 2 \Delta u + \frac{1}{3} u - |u|^{\frac{8}{3}} u =0 \quad \text{in }\Bbb R^3.\tag2$$Let $Q \in H^1(\Bbb R^3)$ be the unique positive solution of (2) anddenote $\beta^*= \frac{b}{2} \| Q \|_{L^2}^{\frac{8}{3}}$. The authorsfirstly investigate the existence and nonexistence of minimizers of (1)by dividing the exponents $p$, $q$ and parameter $\beta$ into five cases.\par The second main result of this paper is devoted to the blow-upanalysis of minimizers when $p=\frac{8}{3}$ and $q \in (\frac{4}{3}, \frac{8}{3} )$. Letting $u_{\beta}$ be anon-negative minimizer of (1) with $\beta < \beta^*$, the authorsshow that the blow-up of $u_{\beta}$ occurs as $\beta \nearrow\beta^*$ and the mass of minimizers concentrates at an inner point of$\Omega$, or the neighborhood of a boundary point. The key of theproof is a refined energy estimate of $e(\beta_k, \lambda)$ as $\beta_k\nearrow \beta^*$.