  We study the quantum phase transition upon variation of the fermionic density
$\nu$ in a solvable model with random Yukawa interactions between $N$ bosons
and $M$ fermions, dubbed the Yukawa-SYK model. We show that there are two
distinct phases in the model: an incompressible state with gapped excitations
and an exotic quantum-critical, non-Fermi liquid state with exponents varying
with $\nu$. We show analytically and numerically that the quantum phase
transition between these two states is first-order, as for some range of $\nu$
the NFL state has a negative compressibility. In the limit $N/M\to \infty$ the
first-order transition gets weaker and asymptotically becomes second-order,
with an exotic quantum-critical behavior. We show that fermions and bosons
display highly unconventional spectral behavior in the transition region.
