You are, perhaps, thinking about a 'negativist' null hypothesis of the general form $\text{H}_{0}^{-}\text{: }|\theta|\ge \Delta$, with $\text{H}_{\text{A}}^{-}\text{: }|\theta|< \Delta$, as used in tests for equivalence, and where $\Delta$ is the smallest effect size that you care about a priori to the test. (As expressed here, the equivalence region ($-\Delta, \Delta$) is symmetric, although it need not be.)
In plain language, for a one-sample or two-sample test the null is "There is an effect of magnitude at least as large as $\Delta$, and the alternative is "The magnitude of the effect is less that $\Delta$." The omnibus form of this null hypothesis would be "There is an effect of magnitude at least $\Delta$ between every group."
Note: For a continuous distribution we cannot simple invert the null hypothesis from the two-sided test for difference (i.e. $\text{H}_{0}^{+}\text{: }\theta = 0$) to be $\text{H}_{0}^{+}\text{: }\theta \ne 0$, because we would have to find (probabilistic) evidence
in favor of $\text{H}_{\text{A}}\text{: }\theta = 0$, but the probability of a continuously distributed variable (e.g., normal, $t$, etc.) exactly equaling a specific number is 0 (i.e. $P(X = c) = 0$), and so you would never reject such an inverted null (i.e. $\text{H}_{0}\text{: }\theta \ne 0$ is no good).
Selected References
Anderson, S., & Hauck, W. W. (1983). A new procedure for testing equivalence in comparative bioavailability and other clinical trials. Communications in Statistics—Theory and Methods, 12(23), 2663–2692.
Reagle, D. P., & Vinod, H. D. (2003). Inference for negativist theory using numerically computed rejection regions. Computational Statistics & Data Analysis, 42(3), 491–512.
Wellek, S. (2010). Testing Statistical Hypotheses of Equivalence and Noninferiority (Second Edition). Chapman and Hall/CRC Press.
See also: [equivalence], equivalence test, [tost]
