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The anti-maximum principle for the homogeneous Dirichlet problem to − Δ p u = λ | u | p − 2 u + f (x) with positive f ∈ L ∞ (Ω) states the existence of a critical value λ f > λ 1 such that any solution of this problem with λ ∈ (λ 1 , λ f) is strictly negative. In this paper, we give a variational upper bound for λ f and study its properties. As an important supplementary result, we investigate the branch of ground state solutions of the considered boundary value problem in (λ 1 , λ 2). [ABSTRACT FROM AUTHOR]

We study the solvability of the Zakharov equation Δ 2 u + (κ − ω 2) Δ u − κ div e − | ∇ u | 2 ∇ u = 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. [ABSTRACT FROM AUTHOR]