Summary: We obtain the critical exponent for the Dirichlet exterior problem \[ \begin{aligned} i u_t + \Delta u & = \lambda | u |^p, \text{ in } (0, \infty) \times D^c, \\ u(t, x) & = f(x), \text{ in } (0, \infty) \times \partial D, \\ u(0, x) & = g(x), \text{ in } D^c, \end{aligned} \] where \(u\) is a complex valued function, \(D = \overline{B(0, 1)}\) is the closed unit ball in \(\mathbb{R}^N\), \(N \geq 3\), \(D^c\) is its complement, \(p > 1\), \(\lambda \in \mathbb{C} \backslash \{0 \}\), \(f \not\equiv 0\), \(f \in L^1(\partial D; \mathbb{C})\) and \(g \in L_{loc}^1(\overline{D^c}; \mathbb{C})\). More precisely, we show that cm

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If \(1 < p < p^\ast : = \frac{N}{N - 2}\) and \[ \operatorname{Re} \lambda \cdot \operatorname{Im} \mathop{\int}\limits_{D^c} g(x) h(x) d x < 0 \text{ and } \operatorname{Re} \lambda \cdot \operatorname{Re} \mathop{\int}\limits_{\partial D} f(x) d S_x < 0 \] or \[ \operatorname{Im} \lambda \cdot \operatorname{Re} \mathop{\int}\limits_{D^c} g(x) h(x) d x > 0 \text{ and } \operatorname{Im} \lambda \cdot \operatorname{Im} \mathop{\int}\limits_{\partial D} f(x) d S_x < 0, \] where \(h\) is a certain harmonic function, then the considered problem possesses no global weak solution.

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If \(p > p^\ast\), then the problem admits global solutions for some \(\lambda\), \(f\) and \(g\).