of all the regions obtained for different values of $\rho_1$ and $\rho_2$ satisfying $\rho_1^2+\rho_2^2\leq 1$. This union is shown in Fig. \[fig:Gaus2\].

Conclusion {#sec:conclusion}
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We considered a three-user cognitive radio network with two primary users and one cognitive user and investigated its capacity region in the strong interference regime. For this purpose, we introduced the three-user Cognitive Interference Channel (C-IFC) by providing cognition capabilities for one of the transmitters in the three-user IFC. We derived two sets of strong interference conditions under which we established the capacity regions. Under these conditions, all three messages are required at all receivers. We also found capacity results for the Gaussian case.

![Capacity region for the Gaussian three-user C-IFC under strong interference conditions $Set_G$.[]{data-label="fig:Gaus2"}](Case2.eps){width="11cm"}

Proofs of Theorem \[thm:ach1\] and Theorem \[thm:ach2\] {#app:ach_proof}
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We propose the following random coding scheme, which contains superposition coding in the encoding part and simultaneous joint decoding in the decoding part. All messages are common to all receivers, i.e., all three receivers decode $m_1$, $m_2$ and $m_3$.

***Codebook Generation**:* Fix $p(.)\in{\mathcal{P}}$. For $u\in\{1,2\}$, generate $2^{nR_{u}}$ i.i.d $x_{u}^n$ sequences, each with probability $\prod\limits_{i=1}^np(x_{u,i})$. Index them as $x_{u}^n(m_{u})$ where $m_{u}\in[1,2^{nR_{u}}]$. For each $(x_{1}^n(m_{1}),x_{2}^n(m_{2}))$, generate $2^{nR_{3}}$ i.i.d $x_{3}^n$ sequences, each with 