Roth-type theorem with unequal powers of Piatetski-Shapiro primes

Let Formula: see text be nonzero integers satisfying Formula: see text. We consider solutions of the unequal powers equation Formula: see text, where Formula: see text and Formula: see text belong to two certain sets Formula: see text and Formula: see text of Piatetski-Shapiro primes corresponding to Formula: see text, respectively. We prove a Roth-type result that for Formula: see text and Formula: see text, whenever the densities of Formula: see text and Formula: see text satisfy a certain lower bound condition, the equation has nontrivial solutions.