  We study the solvability of the Zakharov equation $$\Delta^2 u +
(\kappa-\omega^2)\Delta u - \kappa \,\text{div} \left(e^{-|\nabla u|^2} \nabla
u\right) = 0$$ in a bounded domain under homogeneous Dirichlet or Navier
boundary conditions. This problem is a consequence of the system of equations
derived by Zakharov to model the Langmuir collapse in plasma physics.
Assumptions for the existence and nonexistence of a ground state solution as
well as the multiplicity of solutions are discussed. Moreover, we consider
formal approximations of the Zakharov equation obtained by the Taylor expansion
of the exponential term. We illustrate that the existence and nonexistence
results are substantially different from the corresponding results for the
original problem.
