American Institute of Mathematical Sciences

$\mathcal{M} = { f_{\lambda}(z) = \lambda f(z) : f(z) = \tanh(e^{z}) \mbox{for} z \in \mathbb{C} \mbox{and} \lambda \in \mathbb{R} \setminus \{ 0 \} }$

is studied. We prove that there exists a parameter value $\lambda^$* $\approx -3.2946$ such that the Fatou set of $f_{\lambda}(z)$ is a basin of attraction of a real fixed point for $\lambda > \lambda^$* and, is a parabolic basin corresponding to a real fixed point for $\lambda = \lambda^$*. It is a basin of attraction or a parabolic basin corresponding to a real periodic point of prime period $2$ for $\lambda < \lambda^$*. If $\lambda >\lambda^$*, it is proved that the Fatou set of $f_{\lambda}$ is connected and, is infinitely connected. Consequently, the singleton components are dense in the Julia set of $f_{\lambda}$ for $\lambda >\lambda^$*. If $\lambda \leq \lambda^$*, it is proved that the Fatou set of $f_{\lambda}$ contains infinitely many pre-periodic components and each component of the Fatou set of $f_{\lambda}$ is simply connected. Finally, it is proved that the Lebesgue measure of the Julia set of $f_{\lambda}$ for $\lambda \in \mathbb{R} \setminus \{ 0 \}$ is zero.

$-\Delta u + u - \mu \frac{u}{|x|^2}=|u|^{2^*-2}u + f(u),$
$u \in H^1_r (\R ^N),(1)$

where $2^$*$=\frac{2N}{N-2}$ is the critical Sobolev exponent for the embedding $H^1_r (\R ^N) \rightarrow L^{2^}$*$ (\R ^N)$, $\mu \in [0, \ (\frac {N-2}{2})^2)$ and $f: \R \rightarrow\R $ is a function satisfying some conditions. The main results obtained in this paper are that there exists a nontrivial solution of equation (1) provided $N\ge 4$ and $\mu \in [0, \ (\frac {N-2}{2})^2-1] $ and there exists at least a pair of nontrivial solutions $u^+_k$, $u^-_k$ of problem (1) for each k $\in N \cup \{0\}$ such that both $u^+_k$ and $u^-_k$ possess exactly k nodes provided $N\ge 6$ and $\mu \in [0, \ (\frac {N-2}{2})^2-4]$.