Pseudoconvexity

Remarks

Example: the unit ball $B = \set{z \in \C^n : \abs{z} < 1}$ with $\varphi(z) = -\log(1 - \abs{z}^2)$. The function $\abs{z}^2$ is plurisubharmonic, and $t \mapsto -\log(1 - t)$ is convex and increasing on $[0, 1)$, so $\varphi$ is plurisubharmonic and continuous. The set $\set{\varphi < c}$ is the ball of radius $\sqrt{1 - e^{-c}}$ (empty for $c \le 0$), whose closure is compact and contained in $B$.

Non-example: $D = \set{z \in \C^2 : 0 < \abs{z} < 1}$. Let $\varphi$ be any continuous plurisubharmonic function on $D$, and let $M$ be its maximum on the compact set $\set{\abs{z_1} \le 1/4, \ \abs{z_2} = 1/2} \subseteq D$. For $0 < \abs{t} \le 1/4$, the disc $\zeta \mapsto (t, \zeta/2)$, $\abs{\zeta} \le 1$, lies in $D$, so by subharmonicity $\varphi(t, 0) \le M$. The points $(t, 0)$ accumulate at the origin, which is not in $D$, so $\set{\varphi < M + 1}$ does not have compact closure in $D$, and no exhaustion of the required kind exists.

Equivalent forms: for $D \ne \C^n$, $D$ is pseudoconvex exactly when $-\log d(z)$ is plurisubharmonic on $D$, where $d(z)$ is the distance from $z$ to the boundary. If $D$ has $C^2$ boundary, pseudoconvexity is equivalent to Levi pseudoconvexity: the Levi form of a defining function is positive semidefinite on the complex tangent space at each boundary point.