부산대학교 도서관

This paper concerns the following Kirchhoff equation: $$ \cases - \left( a + b \int_{\Omega} |\nabla u|^2 \,dx \right) \Delta u = \mu u + \beta |u|^{p} u + \lambda |u|^q u, &\quad x\in \Omega,\\ u=0, &\quad x \in \partial \Omega, \endcases $$ 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. We are 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 existence and nonexistence of constraint minimizers of (1) are closely related with 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) and denote $\beta^*= \frac{b}{2} \| Q \|_{L^2}^{\frac{8}{3}}$. The authors firstly 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-up analysis of minimizers when $p=\frac{8}{3}$ and $q \in ( \frac{4}{3}, \frac{8}{3} )$. Letting $u_{\beta}$ be a non-negative minimizer of (1) with $\beta < \beta^*$, the authors show 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 the proof is a refined energy estimate of $e(\beta_k, \lambda)$ as $\beta_k \nearrow \beta^*$.