부산대학교 도서관

This paper deals with the following fractional Kirchhoff-type elliptic problem: $$ \cases -\left(a+b \int_{\Bbb{R}^{N}} \int_{\Bbb{R}^{N}}|u(x)-u(y)|^{2} K(x-y) \, \roman{d} x \, \roman{d} y\right) \Cal{L}_{K} u\\ \qquad\qquad\qquad=|u|^{2_{\alpha}^{*}-2} u+\mu f(u), \quad x \in \Omega,\\ u=0, \quad x \in \Bbb{R}^{N} \setminus \Omega, \endcases $$ where $\Omega \subset \Bbb{R}^{N}$ is a bounded domain with smooth boundary, $\alpha \in(0,1)$, $2 \alpha0$, and $\Cal{L}_{K}$ is a nonlocal operator defined as $$ \Cal{L}_{K} u(x)=\frac{1}{2} \int_{\Bbb{R}^{N}}(u(x+y)+u(x-y)-2 u(x)) K(y) \, \roman{d} y, \quad x \in \Bbb{R}^{N}, $$ for a suitable $K\: \Bbb{R}^{N} \setminus \{0\} \rightarrow(0, \infty)$. In particular, the kernels under consideration include $K(\cdot)=|\cdot|^{-(N+2 \alpha)}$, which is the one of the fractional Laplace operator $$ (-\Delta)^{\alpha} u(x)=-\frac{1}{2} \int_{\Bbb{R}^{N}} \frac{u(x+y)+u(x-y)-2 u(x)}{|y|^{N+2 \alpha}} \, \roman{d} y, \quad x \in \Bbb{R}^N. $$ \par Under suitable conditions on $f$, for $\mu$ large enough, the authors prove the existence of a ground state {\it sign-changing} solution whose energy is strictly larger than twice that of the ground state solutions, by using variational methods and a quantitative deformation lemma.